TITLE: Is modulus operation necessary to define $\mathbb{C}$?
QUESTION [5 upvotes]: In my complex analysis lectures, the following definition (translated) is given for $\mathbb{C}$:

A complex plane $\mathbb{C}$ is a number plane $\mathbb{R}^2$ with the operations of modulus, addition and multiplication as defined before.

EDIT: the modulus operation is defined on $\mathbb{R}^2$: $$(x,y) \in \mathbb{R}^2 \rightarrow |x+iy| = \sqrt{x^2+y^2}. $$
I do not doubt that a properly defined field $\mathbb{C}$ would possess the modulus operation. However, is the modulus operation necessary to properly define $\mathbb{C}$? What if I define $\mathbb{C}$ as

a number plane $\mathbb{R}^2$ with the operations of addition and multiplication as defined before.

Can another field $\mathbb{D}$ possess the same operations of addition and multiplication as $\mathbb{C}$, but a different modulus operation and a different field structure?

REPLY [2 votes]: I am guessing that complex analysis lecturer wants $\mathbb C$ to be more than merely an algebraically closed field with power $2^{\aleph_0}$.  They want also the topological structure given by the modulus.  Note that for the real numbers, we have $x>y$ definable in terms of multiplication and addition, so we can define the topological structure from the field structure.  Not so for the complex numbers.