Let $z \in \mathbb{C} \backslash \{ 0 \}$ and $\alpha \in \mathbb{C}$, with $\log z$ a complex number such that $\exp(\log z) = z$. For what combinations of $\alpha$ and $z$ would the definition $z^\alpha = \exp(\alpha \log z)$ be unambiguous?
And is there also a geometric way to view this answer?
First of all, I'm assuming that your query has typo, and that you intended $\;z^{\alpha} = \exp({\alpha} \log z).$
Assuming that I'm correctly following the convention of Palka ("An Introduction to Complex Function Theory"), and assuming that $\alpha = u + iv$, then
$\exp\{(\alpha \log z)\} = \exp\{(u + iv)(\log |z| + i[\text{Arg} z + 2k\pi])\}$
$ = \exp\{[u \log |z| - v(\text{Arg} z + 2k\pi)] + i[v \log |z| + u(\text{Arg} z + 2k\pi) ] \}.$
As I see it, this will be unambiguous if and only if
(1) $\;\exp\{v(\text{Arg} z + 2k\pi)\}\;$ is unambiguous
and
(2) $\;\exp\{iu(\text{Arg} z + 2k\pi)\}\;$ is unambiguous.
If I'm not mistaken, this will happen if and only if
(1) $\;v = 0$
and
(2) $\;u(2k\pi) \;\equiv\; 0 \pmod{2\pi}\;$ for any integer $\;k$ which implies that
$u\;$ must be an integer.