That’s like saying the standard choice of branch cut for the complex logarithm is arbitrary.
When you complexify, things get messier. My point is that making a generalization is possible (though it’s probably best to sum over integers with 0 \leq arg(z) < \pi, as you pointed out), which is the only claim I’m interested in disputing. Whether it’s nice to look at is irrelevant to whether it’s functional enough to be punnable.
You’re right—the generalization works.
Mainly what I don’t like about it is that \sigma(z) no longer has the nice properties it had over the integers: for example, it’s no longer multiplicative. This doesn’t stop Gaussian integers from being friendly, though, and the rest is a matter of aesthetics.