So what exactly is a complex number?

[Boris Borcic]

Paul Rubin wrote:
> Also, for
> example, the derivative of a complex valued function means something
> considerably stronger than the derivative of a real valued function.

I vaguely remember (it has been decades) an amazingly beautiful russian doll 
system of four of five theorems, the first hypothesizing a function of a complex 
variable with a property that IIRC appeared somewhat weaker than 
differentiability, and the latter theorems not hypothesizing anything more, but 
pushing the consequence further to ever stronger properties.