I don't think that's correct. You can also rotate around the direction you are looking. Rotations in 2D are one-dimensional. Complex numbers are two-dimensiona

I don't think that's correct. You can also rotate around the direction you are looking. Rotations in 2D are one-dimensional. Complex numbers are two-dimensional but we fix one degree of freedom by fixing the length. Rotations in 3D are there-dimensional. Quaternions are four-dimensional but we fix the length again.