# $4D$ rotations and quaternions

I have a question about $4D$ rotation:
I programmed a little $4D$ game and I used the classical hyper-sphere coordinates, to rotate a vector.

It works, but it has some problems:
(just for clarity I take the cartesian coordinates, translate in hypersphere coordinates, add the angle I wish than translate back to cartesian)
This procedure causes jumble locks and approximation errors.

I'm asking you if it is possible like in $3D$ to have quaterinions to manage the rotations?

I'll prefer not to use rotation matrix, which has other problems:

Thanks in advance for the help!
If you want to have a look at the game: