Surjectivity of the complex exponential without using $π$

Question

I want to prove that the exponential $\exp\colon ℂ → ℂ^×$ is surjective without using polar coordinates and without even using a definition of $π$. Is there such a conceptional argument?

Say, I already know that $\exp(ℝ) = (0..∞)$, and that $\exp$ is a continuous homomorphism commuting with complex conjugation. Then it suffices to show $\exp(iℝ) = S^1$ of which I then at least know that $\exp(iℝ) ⊂ S^1$ is a path-connected subset. How can I proceed?

Answer 1

$\exp(i\mathbb R)$ is a connected subgroup of $S^1$. There aren't too many of those...