4-dimensions to a 3-dimensional game would be like FEZ’s 3-dimensions are to its 2-dimensional gameplay.
I’ve toyed with the idea too, and what I keep coming back to is the FEZ solution, that is, to have the “extra” axis rotate in steps, so as not to have to render all the possible intermediate states.
To follow up the FEZ example, in that game the third axis rotates in 90º increments, which results in 4 different 2-dimensional versions of the map to play with (the top and bottom faces of the level are ignored).
If the axis could be moved to an arbitrary position, it would result in a virtually infinite number of versions of the map to move about, which would make gameplay design unmanageable, and possibly extremely confusing.
My guess is that in a 3-d/4-d game, you’d have a larger number of 3-d versions of the map to visit when rotating, and the trick would be to have entities (for example, enemies) moving about in a way that makes sense regarding the “face” of the hypercube they’re still on.

I love 4-dimensional geometry and its weirdness. Hope you get somewhere interesting with your project.