Complex Numbers cannot be Ordered Compatibly with Ring Structure

Theorem
Let $\struct {\C, +, \times}$ be the field of complex numbers.

There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$.

Also presented as
Some sources word this result in the following form:
 * As applied to complex numbers, the phrases "greater than" or "less than" have no meaning. Inequalities can only occur in relations between the moduli of complex numbers.

This tells only part of the story.

By the Well-Ordering Theorem, it is possible to apply an ordering of some kind upon $\C$.

Unfortunately, as is proven here, such an ordering would not actually be compatible with the structure of $\struct {\C, +, \times}$, and so of little practical use.