= Solution
For $z=re^{i\theta}\ne0$, fix one argument $\theta$. Every value of the <complex logarithm> is $\log r+i(\theta+2\pi k)$, $k\in\mathbb Z$. By <complex exponentiation>, $z^z=\exp[z(\log r+i(\theta+2\pi k))]$. Taking the real part of its exponent gives all possible <moduli>:
$$
\boxed{|z^z|=\exp\!\left[r\cos\theta\log r-r\sin\theta(\theta+2\pi k)\right],\qquad k\in\mathbb Z.}
$$
Equivalently these are $r^{\operatorname{Re}z}e^{-\operatorname{Im}z(\theta+2\pi k)}$. If $z$ is real and nonzero the modulus is the single value $r^{\operatorname{Re}z}$; otherwise the values form a geometric progression with ratio $e^{-2\pi\operatorname{Im}z}$. Changing the initially chosen argument just reindexes the same set. The logarithmic definition excludes $z=0$. This is the <modulus of a complex self-power>.
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