Complex numbers are 2D. It is a commonly sited result that there is no 3D or 4D analogue of the complex numbers.
I just want to be clear on exactly what this result says:
It is impossible to construct a 4D field.
It is impossible to construct a 4D field which the complex numbers are a subset of.
It is impossible to construct a 4D field which the real numbers are a subset of.
(I'd also like to know why it's true - but given that I didn't understand the answer the first $2^7$ times, I doubt I'll understand it this time either. :-} Oh well...)