Isn't the prevailing wisdom to skip 3-space and go directly to 4-space, which makes things even simpler and more powerful -- note 4D projective space (homogeneous coordinates, not quaternions and not counting a time dimension).

Reasons for going to 4D are numerous and varied, such as symplectic geometry only works in even dimensions, image/video reconstruction, matrix multiplication, transformations, and other reasons related to computer vision, gauge theory, and topology.