How to show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$?

Question

How can I show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$? I can see that it suffices to work with the unit disk, i.e. it suffices to show that $|z|=1$ for all $z\in D(0,1)$, but how should I go about showing this? Any hints would be appreciated.

Answer 1

1. $|i|=1^{1/4}=1$.
2. $|e^{i\theta}|<=|cos\theta|+|i\sin\theta|\leq2$.
3. If for some $\theta$ we have $|e^{i\theta}|\neq 1$, then there exist $n\in\mathbb{Z}$ such that $|e^{i\theta}|^n>2$. But then $|e^{i n \theta}|>2$, contradiction.