(This is probably actually not the best way to go about things, and is half in jest. You'll see my point at the end.)

Alright, so I decided to try to extend the divisor function σ(n) to the Gaussian Integers. I realized that the best way to do this is to make it so that only certain numbers can be divisors, because making them all divisors makes the divisor function equal to 0. After much trial and error, I figured out what would make the best sections. The two sections are:

Section 1: All numbers in the first and second quadrants of the Cartesian Plane, including the negative reals and not including the positive reals.

Section 2: All numbers in the third and fourth quadrants, including the positive reals and not including the negative reals.

All of the divisors of a Gaussian Integer Z are defined as the Gaussian Integers x in the same section as Z, that, when multiplied by another Gaussian Integer y in the same section, equal Z.

Using this definition, we can find the value of the divisor function for a few different complex numbers:

σ(i) = 0

σ(1 + i) = 0 (I think)

σ(-2i) = (-2i) + (1) + (1 - i) + (2) + (-i) = 4 - 4i

σ(3i) = 0

Wait, let's look at the abundancy at a couple of those. i and 3i both have an abundancy of 0. That means they are friendly. You might not notice, but this has huge ramifications.

...I just proved the existence of imaginary friends.

(Yes, this was inspired by

this xkcd)