Is it possible and is it a good idea to introduce the additive group and metric space $\mathbb{C}/2i\pi \mathbb{Z}$ very soon, at the same time as the complex logarithm $\log(r e^{i \theta}) = \ln(r)+i \theta, \theta \in (-\pi,\pi], r > 0$ ?
With $\log : \mathbb{C}^* \to \mathbb{C}/2i\pi \mathbb{Z}$ we can say $\log(ab) = \log(a)+\log(b), \log(a^n) = n \log(a)$, $\prod_n (1+a_n)$ converges iff $\sum_n \log(1+a_n)$ converges.