But they wouldn't extend to 4D. You can treat complex numbers as 2D rotors (a 2D rotor is cos alpha + xy sin alpha, so xy can be replaced with i) and quaternio

But they wouldn't extend to 4D. You can treat complex numbers as 2D rotors (a 2D rotor is cos alpha + xy sin alpha, so xy can be replaced with i) and quaternions as 3D rotors. But if you build Cayley numbers from quaternions they don't have the same geometric meaning as 4D rotors.