A normal person (mathematician excluded) cannot visualize n-dimensional very easily (or at all). Computers can easily compute the necessary math.

Clearly the display of that "extra" dimensional information is the problem. To remedy this the level will be a true 4-d hyperspace (w,x,y,z) but at any time t the quake 3 map will be a "slice" of that 4-d hyperspace as it intersects a 3-d space.

This is much like showing a sphere to a flatlander by pusing it through her flat 2-d universe. It would first appear as a point, then as a circle growing in size, then shrinking, and finally disappearing altogether. Now just extend that analogy up a dimension. You'll have a quake 3 map that confusingly changes shape as time progresses. It even presents the possibility that a player could become "trapped" in an area of the map that is shrinking out of existence.