For simplicity, my rendering engine is a dead-simple raytracer which doesn't even bother with reflections- just cast rays and record the first surface they run into. Handling rotations and projections is really easy, but there are two problems I have found:
1. It's just really frickin' slow. Raytracing just ain't fast enough, though I may be able to find some additional optimizations to make it better.
2. In terms of gameplay, navigating the space is really, really hard. It is so easy to get completely lost regarding which direction you're facing and what hyperplane you're on when you can make arbitrary rotations in all four dimensions. It might be something I could get used to, but so far I feel like it's practically necessary to have some mechanism that will snap you back into alignment with some set of gridlines.
EX: walls in the way. <Set bomb off> Walls gone but cave collapses. <walk though wall> <go back in time> <keep walking>.
Or simply let you flip times direction. Going vary slow or backward prevents changing things, but you get to move. So, you might be able to for example walk on water when moving backward in time etc.
It's a different concept than true 4D though.
My favorite 2048 variation was the 4D version. It was trippy but I felt like I got my head around that. I think it was because the game rules were so simple that instead of trying to visualize what was going on, my brain just accepted the patterns.
We see in 2 dimensions and we infer the third from various cues. (The quote that we are "dimly aware of a fourth", which you may have heard, refers to time and is a different issue than a fourth spatial dimension.)
We have three dimensions, all orthogonal to each other. We can translate, rotate, and project things in between them.
So all we can do is apply those same principles and see what comes out - we shouldn't expect it to match anything recognizeable though.
xy, xz, etc. are not planes in 4d space - they are 3d-hyperplanes, and perpendicular to them are planes, not axes. Imagine this: you have a 4d vector and you hold x and y constant. You still have 2 degrees of freedom: a plane, not a line.