How do I formally define complex exponentiation?

The only such quotient of $\mathbb{C}^*$ is the trivial, one-element group (NB, we don't look for a quotient of $\mathbb{C}$, essentially because of the multiplicative behavior of $\exp$).

For a fixed base $z \in \mathbb{C}$ and any choice of branch $\log$ of the logarithm function, we can define a branch of the exponential function with that base by $$f(w) := \exp(w \log z).$$ As usual, any other branch $\widetilde{\log}$ of the logarithm function can be written as $$\widetilde{\log}(\zeta) := \log \zeta + 2 \pi i g(k)$$ for some integer-valued (and not necessarily continuous) function $g$.

The corresponding branch of the exponential function with base $z$ is \begin{align} \widetilde{f}(w) &:= \exp(w \widetilde{\log} z)\\ & = \exp(w (\log z + 2 \pi i g(w)))\\ & = \exp(w \log z) \exp(2 \pi i g(w) w)\\ & = f(w) \exp(2 \pi i g(w) w) \textrm{.} \end{align}

We can now formalize the question: We are looking for a subgroup $\Gamma \subset (\mathbb{C}^*, •)$ such that the composition $$\mathbb{C} \stackrel{f}{\to} \mathbb{C}^* \to \mathbb{C}^* / \Gamma$$ is independent of the choice of branch $f$ of the exponential function with the base $b$, or equivalently, of the choice of branch $\log$ of the logarithm. (The second map in the composition is just the canonical quotient map.)

Now, by the above computation, $\Gamma$ must contain every possible value of $f(w)^{-1} \widetilde{f}(w)$, that is, it must contain $$\exp(2 \pi i k w)$$ for every value of $w$ and every integer $k$, but the set of such values is the image of $w \mapsto \exp(2 \pi i w)$, which is $\mathbb{C}^*$ itself. Thus, the only admissible map $\mathbb{C}^* \to \mathbb{C}^* / \Gamma$ is the trivial map $\mathbb{C}^* \to \{ 1 \}$ itself.

Of course, there is still a way to define complex exponentiation formally---this is precisely the definition of $f(w)$ in the first display equation above.

For variety, we could use a Riemann surface: the set of all points $(z,w)$ such that $\exp(w) = z$.

The points of this surface should be thought of as follows: the first coordinate expresses a point on the complex plane, and the second coordinate expresses additional information: a choice of the branch of the logarithm.

Let $\hat{z} = (z,w)$. The multi-valued function $\log z$ on the complex plane lifts to a well-defined, continuous, and differentiable function $\log \hat{z}$ on the Riemann surface above, given by $\log \hat{z} = w$, and this function satisfies $\exp(\log \hat{z}) = z$.

Furthermore, we can define $\hat{z}^a = \exp(a \log \hat{z})$ to get a well-defined exponentiation function everywhere defined on the Riemann surface.

We could actually make $\exp$ take values on your Riemann surface, so that $E(a) = (\exp(a), a)$. Then $\log$ and $E$ are inverse functions. The corresponding version of complex exponentiation is $\hat{z}^a = (\exp(aw), aw)$.