I think of it as the opposite: there exists a Lie group of spatial rotations, so we want to use it, and we found a relatively convenient notation. There are ma

I think of it as the opposite: there exists a Lie group of spatial rotations, so we want to use it, and we found a relatively convenient notation. There are many other kinds of 'complex' numbers (http://en.wikipedia.org/wiki/Hypercomplex_number http://en.wikipedia.org/wiki/Hypercomplex_number) - but you probably won't hear about them outside of mathematics and physics because they're less useful. The same applies to 2d - the SO(2) group exists; we map it to the complex numbers because it's convenient to do math with.