I'd say Quaternions are necessary purely because they avoid gimbal locking common with 3 axis Euler rotations.

I know in Blender all other rotation options are really just quaterinions under the hood. But an advantage Euler rotations have over quaternions is continuous rotation. I can set 2 key frames with linear extrapolation and have them just continue along each axis forever. Everything quaternion ideally should be bounded to the 4d unit circle.