This puzzle by Andrey Astrelin supports a large variety of twisty puzzles in four-dimensional space and above. Puzzles are described in text file, so anybody who understands its syntax can add their own puzzles. You define symmetry group, rotational axes, cutting planes, possible twists and body boundary - and your own puzzle is ready :)
But even without that there are more than two dozens very different puzzles that include all six regular polytopes (face, edge, and vertex-turning tesseracts), a vertex-turning 5D cube, three 6D puzzles (a duo-tetrahedron and two triaprisms) and others.
The user interface for cell-turning 4D puzzles is very simple - just click for face twists or macro application. For other kinds of twist axes it is slightly more tricky.
Download the puzzle here. Unpack the zip-file in new folder and run MPUlt.exe. This program requires Microsoft .NET 3.5 and DirectX
Intructions for the program controls are here.
The source code is here. Use it as you like so long as you give clear and obvious credit to the author.
Magic Puzzle Ultimate took many ideas from the MagicCube4D project. I would like to thank its developers Don Hatch, Melinda Green, Jay Berkenbilt and Roice Nelson for this wonderfull puzzle and for the very interesting Hypercubing discussion group!
One cell of 24-cell
Half-cut tesseract after one twist
32-cell - dual to rectified tesseract
Scrambled vertex-turn 5D cube
6D triaprism after some twists
One cell of triaprism after one twist