What is the generalized definition of '<' and '>' for complex numbers?

Question

I'd expect this question to be asked here before, but I've not been able to find it.

The generalized definition of the multiplication operator for complex numbers is simple:

The product of the lengths and the sum of the angles (relatively to the $X$-axis).

But what is the generalized definition of the $<$ and $>$ comparators for complex numbers?

Answer 1

There isn't one. There is no ordering compatible with the algebraic structure as there is for the case of real numbers. (Eg because everything is a square.)

Occasionally it is useful shorthand to write $z>0$ or something similar when referring to a condition on a complex number $z$, but this means that $z$ is real and greater than 0.

Answer 2

One interesting partial order on $\mathbb{C}$ is the following: $z \leq z'$ if $| z | = 0$ or ($| z | \leq | z' |$ and $\arg z = \arg z'$).