While n-dimensional Quake, in theory, would be an extremely interesting thing, the fact of the matter is that not everyone gets the same thrill out of watching several bots destroy each other in new and interesting ways. The true problem lies at the heart of the matter: expanding the bot-match to include the ever-limited perspective of the actual Quake player.
It took me a few seconds to get used to thinking in two dimensions, back when Wolfenstein 3D was released. The Doom series introduced an entirely new challenge: psuedo-3D gaming. This tiny shift took a couple of hours to understand, and I found myself reeling at the concept of how hard it would be to actually play a game where vertical aiming was not handled by the computer.
Unfortunately, I saw the enemy and it was Quake. I spent days trying to understand the concept of actually thinking, moving, looking, and aiming in three dimensions. I spent months working on coordinating the efforts of the hand I kept on the keyboard with the hand I used to move the devilish mouse. Even after a few years, I'm still struggling to cope with what has become a staple in the first person shooter community.
My point is that even if you could strap a volunteer into a rig that is designed for n-dimensional movement, from conventional compass directions with the left hand to twelfth dimension, uh, movements with an optical sensor designed to pick up the miniscule flaring motions of the nostrils, there are very few humans on earth that would have the sheer will and mental capabilities required to grasp both the physics of the game and comprehend the controls. While it would be interesting to think that opening and closing your jaw could extend your digital self into dimensions normally only seriously discussed on Star Trek, the fact that sneezing could inadvertantly land your player in ancient Greece would tend to put a damper on your search for the enemy flag.