Let $\pi$ be a Gaussian prime. Like $p$ is the natural choice for the base in the $p$-adic absolute value $|x|_p = p^{-v_p(x)}$ due to the Product Formula, what is the natural choice for the base $c$ in the $\pi$-adic absolute values $|x|_\pi = c^{-v_\pi(x)}$ on $\mathbb Q(i)$, in each of the 3 cases (up to associates):
- $\pi = 1 + i$
- $\pi \equiv 3 \mod 4$ is a rational prime.
- $\pi = x + iy$ or $x - iy$, where $p = (x + iy)(x - iy) \equiv 1 \mod 4$ is a rational prime?