I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case:
$$ \log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$
For the imaginary integers $\mathbb{Z}[i]$ there's no analogue of factorial, but we can ask for the product of numbers in a given rectangular region:
$$ (M+iN)!_{\mathbb{Z}[i]} = \prod_{0 < m \leq M;0 < n \leq N} (m+in)$$
Does this function already exist under a different name? How do we estimate the phase and magnitude?