Allow $\log : \mathbb{C} \rightarrow \mathbb{C}$ to be multivalued, by which I really mean that it's a relation $\mathbb{C} \rightarrow \mathbb{C}$. We can define $a^b = \exp(b \log a)$ without encountering any real problems (pun intended). This defines a relation $\mathbb{C} \times \mathbb{C} \rightarrow \mathbb{C}$. So for $a,b \in \mathbb{C}$, we're thinking of $a^b$ as being a set of complex numbers.
Request. I'd like to see a comprehensive discussion of the structure of the set $a^b$ for different values of $a$ and $b$. Please make your answers as elementary as possible, so I can offer a link to this question to young people (e.g. 18-20 years of age) and have it be useful to them. Further discussion of the properties of $a^b$ highly appreciated, as long as it's elementary. No branch cuts please, except for brief digressions explaining why e.g. calculus with fully-multivalued $a^b$ is messy, that kind of thing.
Remark. This "maximally multivalued" version of $a^b$ appears in the Gelfond-Schneider theorem.