This webpage is all about 4D objects, which are called polychora.
Polychora are very similar to 3D objects (polyhedra) and lots of things about polyhedra also apply to polychora.
There are 4D equivalents of Platonic solids (see link below to learn what they are), and they are very similar, with ony a few minor differences.
Platonic solids wikipedia page
The biggest difference between the Platonic solids and the 4D Platonic solids is that there are 6 of them in 4D, whereas there are only 5 in 3D.
Prisms can be a lot more interesting in 4D, like having a sphere prism or a rectangular prism prism.
Another interesting thing about 4 dimensions is that you can rotate a 3D object in the 4th dimension to flip its cardinality.
You may have seen a gif or video similar to the one i have below, this specific polychoron is the 4D equivalent of a cube, and it is called a tesseract, and the reason it is moving is because we would percieve stationary 4D objects as a single 3D slice of the whole object, and this tesseract would just look like a cube.
This object does not actually intersect itself, it just appears that way because this is the best way for us to percieve 4 dimensional rotation.
If you look closely at the gif, you can see the 7 of the 8 individual cubic cells that make up the tesseract when it looks like a cube inside a cube, and all 8 of the cells when it looks like 2 cubes attached to each other with an octahedron with the tips cut off surrounding them.
footnote: all images are clickable, and will open the wikipedia page about the object(s) in the image in a new tab.