Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number can be thought of as representing the same thing), I think this way of thinking misses that we also have a field of complex numbers. Point being R^2 isn't a field but C is.

If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).