"Ordering" of Complex Plane

You can of course put a total order on the complex numbers (an example easier than your space-filling curve: simply take the lexicographic order on $\mathbb{C}\cong\mathbb{R}^2$). However, if you are studying the complex numbers as a field, then you'd like your order to be in some sense compatible with sum and multiplication. For example, you want

• if $a<b$, then $a+c<b+c$
• if $a<b$ and $c>0$, then $ac<bc$

and some others. However, this is impossible to obtain in $\mathbb{C}$. Sadly, I don't know the proof of this fact. I would be interested in seeing it though.

You can definitely put an order $\prec$ on the complex numbers. In fact, the order can be a well-order, by the well ordering theorem. The question is whether such an order "is useful."

Here are some things we probably want in the order $\prec$:

• if $0 \prec \alpha, \beta$, then $0 \prec \alpha \beta$.
• if $\alpha, \beta, z, w$ are complex with $\alpha \prec \beta$ and $z \prec w$, then $\alpha + x \prec \beta + y$.

In fact, an order on a field (such as the complex numbers) that satisfies these properties gives us an ordered field. See Ross Millikan's comment for why you cannot put such an order on $\mathbb C$.