Importantly, however, the complex numbers have no total ordering that respects addition and multiplication. In other words, there's no large set of "positive complex numbers" closed under both operations.
This is also the reason why the math in this XKCD strip doesn't actually work.
You can still find divisors for Gaussian integers. If x, y, and xy are all Gaussian integers, which will be trivially fulfilled for any x when y=1, then x, y both divide xy.
You can then generalize the \sigma function by summing over all the divisors of z and dividing by |z|.
The resulting number \sigma(z) lies in C (or maybe Q + iQ), not just Q, but it's perfectly well defined.
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