n-dimensional quake (and other FPS games) would be pretty pointless. Nothing would ever happen!

Quake doesn't really use 3 dimensions. Most of the time you're on a surface; occasionally you get to go from one surface to another. Even various "flying" maneuvers are limited in time and distance, not to mention having a preferred direction (known as "down" by the primitive beings who inhabit planet Earth; note that this preferred direction changes, but always exists). Airplanes have a slightly better use of the third dimension, but even they tend to move in "layers".

If players move along random walks (or Brownian motion), something amazing happens. In two dimensions (or, of course, 1), any two random walks will (almost surely -- and read the definition of "almost surely"!) come arbitrarily close infinitely many times, and indeed intersect. This also applies to random walks on other infinite two-dimensional manifolds; topology has nothing to do here. Since, as mentioned above, Quake is really "two-and-a-half dimensional", it fits into this category. So randomly-moving players in Quake find each other. When they do, things get interesting.

In 3 or more dimensions, random walks diverge. Rapidly. There are (almost surely) only finitely many (and, indeed, very few) approaches between the walks. Lacking radar with unlimited range, in n-dimensional quake, players will never find each other! They'll just wander around, shooting off their weapons into thin air. In fact, they'll never find most of the weapons or ammo, so they're pretty much stuck with what they've got initially.

This is not due to their limited imagination (no matter how much we despise the puny humans who infest this planet), but due to unavoidable geometric properties.

In technical terms, n≥3, n-dimensional quake is boring.