The one application I know for 4D collision detection is 3D continuous collision detection. If two 3D objects are moving, you can parameterize their movement through time to obtain a 4D shape. If those movements are only translational, and the two shapes are convex and have a support function, then testing for contact is easy (e.g. using GJK). However if those movements contain rotations, the resulting 4D shape will be concave so interference detection may be hard.

I do not think anybody actually uses this method though.