This page is about creatures or intelligent beings native to the Fourth Dimension, particularly the spatial interpretation. Read that page too, as the basics it gives you on how the next dimension up works may prove helpful in explaining why the Fourth-Dimensional Lifeform can do what it can.
We Live In A Fishbowl. Let's hope the cat doesn't notice us.
That is to say, our three-dimensional universe occupies only a tiny sliver, infinitely thin, within the greater fourth-dimensional context. For the sake of this page (and the fun implications for gaming), we'll assume that there are life-forms that exist naturally in four-dimensions. We may be beneath their notice, thinner and less pesty than the tiniest ant in comparison to them. Or, they may turn their microscopes upon us, and mess with our reality. We may be more noticeable than we imagine as well, being as obvious to them as man-sized paper-thin silhouettes pressed between two mile-wide planes of glass would be to us.
So one day, they may take note and decide to intrude on our reality. If we're lucky, they may be smart and compassionate. Alternately, they might regard us the way a cat does a goldfish in a bowl.
Either way, by virtue of being fourth-dimensional, it's going to have some creepy powers over us.
The fourth-dimensional lifeform has a better vantage point. At the very least, it can rotate around our little slice of reality in the same way a cat can maneuver around the outside of a fishbowl to get a better look at the helpless little swimmer. Now give that cat X-Ray Vision, and you'll begin to approximate the way in which a 4D critter will be able to spy on us. It can see around corners. It can look through walls. It can see what's in your belly.  
Even if it's got no greater intelligence than a germ, it's likely to sense us before we detect it.
If there's a 4D thing coming after you, don't rely on walls or obstacles to protect you. You can run, but you can't hide, and you can't take shelter. Three-dimensional prey trying to flee from a fourth-dimensional predator is going to have a very hard time.
This isn't quite the same as Teleportation, but it may feel like it. It will have a few restrictions on the 4D movement that might save you:
- It won't be as fast as teleportation. It can pop in and out of our reality pretty quickly, but still has to move through the 4th Dimension to get to the vantage point from which it will step back into our reality. On open terrain, or sliding down an embankment, you'll be able to keep pace with it - unless it's also a speed-based predator back home, or has a fourth-dimensional motorcycle, or some similar speed boost.
- The main advantage it has is the ability to sidestep three-dimensional obstacles and hazards. While it wouldn't be able to materialize inside solid matter, but it would be able to step out of our reality, move beyond where a 3D wall is, then reenter our reality on the other side.
- If fourth-dimensional space is made up of infinite sheets of tightly-packed parallel 3D universes, then this movement may be slowed significantly. It may have to move through hundreds of parallel worlds to find one without the wall that you just "took shelter" behind. As it moves through these parallel worlds, it may become distracted or choose to dine on a parallel you instead of returning to our universe.
- If, on the other hand, 4D space is as empty as our 3D universe, then there may be a large void between us and the next nearest 3D world. in that case a shift out of our reality and back again is like leaning or sidestepping to it, much easier than using the door or climbing a cliff. It could choose to move 50 feet to the door, a foot through it, and then 50 feet back to where you are. But why not just lean an inch perpendicular to our plane, then a foot parallel to our plane, and an inch back perpendicular to just suddenly appear between you and the wall?
Should you try to stand your ground and fight it, it won't be easy. It has the ability to remove itself from the 3-D frame of reference. Probably pretty easy, as instinctive as a dodge. A bullet may be fast enough to hit it, but if it understands the significance of you drawing a gun, it'll just vanish and watch from outside our plane until you let your guard down. See Fair Escape.
There is some small chance that if you can get it into a freefall, it won't be able to move into or out of our reality. Maybe. Assuming the freefall extends to it's whole body, the majority of which will already be outside our reality. Not really the sort of thing you want to test by jumping out of an airplane without a parachute while a monster from beyond is chasing you.
Variable Size and Mass, possibly even Shape-Shifting
To us, the fourth dimensional being will have variable size, and thus variable mass. The 4D character won't perceive any change, it's just a matter of how much of it's body extends into the 3D frame of reference. There's a number of tricks this allows:
- By reaching just a couple limbs or digits into our reality, it may give the impression of being more than one creature. We see half a dozen wormlike creatures intruding into our reality, but they're actually just its fingers. 
- What you're seeing, by the way, is just a 3D cross-section of the creatures body.   Or maybe even a cross section of its' clothing. Depending on the size of the thing in the fourth dimension, it may be able to radically change it's appearance and coloration, which you'd perceive as a limited form of shapeshifting.  It may or may not be aware of what is visible to you, depending on how clearly the boundaries of our three dimensional space are demarcated from its point of view.
- Assuming it's intelligent and has knowledge of its own anatomy, it can keep its vital organs outside our reality, making a critical hit impossible. Our weapons may pass right through it from our point of view, looking like a mighty cleave, while making only a shallow wound as far as the creature is concerned.
- It was pointed out that it can see inside us. It can probably reach inside us as well. It may get automatic critical hits.
- Even an unintelligent 4D beast might be lucky enough to have the metaphysical blubber to be Immune To Bullets, especially if it's body extends a great distance in the fourth spatial dimension.
- If tool-using, it may only extend weapons or probes into our reality. Especially if it hunts 3D prey, what we encounter might in truth be no more than the point of a spear or fishhook.
- Likewise, it can "shrink down" to an unnoticeable size (from our point of view) for stealth purposes. It may have enough energy to exert great force even when manifesting only a tiny presence in our reality. Or, it may be limited to only having the force derived from the portion of it's mass that is in our frame of reference. The GM has some flexibility here: either it gets tiny but is weakened (so it will choose to manifest as largely as is practical), or it has a tiny stealth mode (like pushing with just your fingertip) where it can pull all kinds of poltergeist nonsense.
Plucked from our own Dimensions.
The 4th Dimensional being may have the ability to yank us out of our frame of reference and out into the greater reality. This could yank us into parallel worlds, or just a big void outside our universe. An angry 4D Being might maroon us in a radically different environment. Our senses won't be adapted to it, and we'll have a hard time figuring out what we're looking at.  
It may even throw things at us, knocking us out our frame of reference. Or it could anchor us, leaving us rooted between worlds or fixed in space. And what if it just exhales steadily into our 3D space? Can it blow us loose without ever setting foot in our reality? Can it scramble our organs with a trans-dimensional bronx cheer?
Let's say the 4D being is kind enough to put us back. There's still some ways this could mess you up.
- It can pluck people and things out of our dimensional plane, flip them, and put them back. If it did this to you, your left-right symmetry would be reversed. Your face would look just a little different ("Is that a new hairdo?"), your heart would now be on the right side of your chest, and your handedness would change. 
- Whether or not the flipping of the right and left hemispheres of your brain has other effects is pretty much up to the GM. You could ignore it, or it could justify some major personality alterations.
- It's also completely possible that doing this could simply convert things into antimatter. In other words, you'd better be in vacuum when it does that.
- Rather than a full flip, it's possible it might be able to just tilt you out of sync with our reality. You'd be facing handicaps that were inversely related to the fourth dimensional beings advantages. You might be split between multiple worlds, each cross-section facing a different parallel reality.
Fun With Forces.
The fourth dimensional being may be able to bend or ignore various forces and properties of the natural world. It may resist gravity, electromagnetism, and/or the strong nuclear force or weak nuclear forces. For an explanation of why, see Riemann's Bookworms.
If light is is a fourth-dimensional vibration, then it's possible the 4D being might appear as if it were made of light, or it might emit light with every motion it makes. This would be consistent with theories that it takes a lot of energy to breach dimensions. While I mentioned the ease with which a 4D critter can move into and out of our reality, there is the high likelihood doing so will take enormous amounts of energy. It's 4th Dimensional nature may give it that power, or it could be an extreme effort for it to interact with our reality.
If light is instead cast from some higher dimension, or cast from a different source withing the 4th Dimension, then we can assume a 4D object will cast a 3D shadow over three-dimensional space. In fact one way scientists attempt to visualize 4D objects is by contemplating their 3D shadows. 
Because of how vanishing and reappearing dodges 3D understanding of Causality, it may seem that the 4D creature is a Time Traveler, moving into the future when it's actually just stepping outside our reality. Relativity and Spacetime are full of weirdness like that. 
It's also possible to have a creature that moves freely in the Temporal Dimension instead of (or in addition to) a spatial dimension. Such a creature would combine the advantages listed above with genuine Time Travel, for a truly superior foe that kills you before you're even aware of it. One possible temporal trick (one really only available to the fourth-dimensional lifeform) deserves a page of it's own: Rotating Into Time.
Another notion involves parallel universes and branching time. Let's assume the model of temporal physics where every action or decision creates multiple parallel worlds, one for each possible outcome. The Fourth-Dimensional Lifeform may drift between them freely, and it's actions may create such alternate worlds. In fact, it may have an infinite food supply made up of a single person (or a specific animal) that it likes to eat. Each time it snatches up that meal, it creates another world just like that one, except where the favorite meal hasn't been taken yet. If you think about it, our world could be just the latest variation on another - one that's identical except it's version of you is now a missing persons report. Meanwhile, just out of phase with our reality, the thing that eats only you is napping. Should it awake and grow hungry, it will hunt and devour you. Doing so will create a new reality, where a duplicate you exists, who's read this paragraph, and laughed it off, and another world with a duplicate you who read this paragraph and started trying to figure out how to fight back against the fourth-dimensional horror. Which you do you want to be?
Trying to anticipate the actions of a fourth-dimensional foe will be mind-bogglingly difficult, testing the limits of even a battle-hardened warrior with a degree in astrophysics. It's enough to drive you batty. See Alien Geometries and Go Mad From The Revelation for more ideas on how unnerving it would be to be hunted by a time-traveling dimension-hopping Eldritch Abomination. 
Since very few resources exist that give serious thought to what the ramifications would be of a 4th Dimensional predator, I've had to extrapolate a bit. I feel pretty confident that I didn't mess anything up, but I should mention that I don't have a degree in physics. The parts I extrapolated are mostly couched in terms like "it may be possible", or "if light works this way", so that I don't pass off my own conjectures as certainty or scientific theory. Things I've read as bonafide scientific theory are phrased (above) more like "it can" or "you can't", more definitive than my own speculation, even in cases of theoretical things that haven't been proven yet. I also applied footnotes where appropriate. In other words, if a scientist tells you something on this page is just wrong, be inclined to believe them over me. If they claim something is wrong that I foot-noted, it may be time for someone (me? you? them?) to read up on the subject. Luckily, Hyperspace and Flatland are both quite enjoyable reads as well as being highly informative.
Since creating this page, I've learned of a comic book that featured a Fourth-Dimensional Villain in one issue, and which uses several of these same concepts. Online version here. It may prove useful for helping you visualize some of these ideas - and rest assured your fourth-dimensional lifeform doesn't have to look like a pile of onion rings with eyeballs.
Game and Story Use
- To make a Monster or Alien really creepy or, well, alien, make it fourth dimensional.
- Shadow People might be the sentient shadows of greater 4-D creatures that look down upon our reality, but never enter it.
- Alien Greys might be 4th Dimensional. It could explain UFOs that vary in size or vanish suddenly. They'd be able to abduct you without a trace, and you'd be powerless to move against them. See We Live In A Petri-Dish.
- In Trail of Cthulhu, it is suggested that the Eldritch Abominations, or at least the really nasty ones, are fourth (or higher) dimensional beings. Not all of Great Cthulhu is shaped like a cross between an octopus and a dragon, just the part of him that passes through three-dimensional space is. See also Alien Geometries.
- In a light-hearted and quirky game, the gods might be 4-D. See the Hypersphere page for ideas on how fourth-dimensional geometry could result in the gods unearthly charisma and appearance.
- If used in a setting with Lumineferous Aether, the fourth dimensional lifeform has a few extra tricks it might be able to use, such as bending, creating, or capturing light, change the colors of things, creating reflections and holograms, etc. Of course, that's small potatoes compared to what a 4-D lifeform can already do without the aether, but it's still kinda cool.
- Combine a 4-D predator with Quantum Immortality for a bleak horror tale (or an interesting/surreal puzzler reminiscent of Aeon Flux) where you keep getting eaten again and again.