Essentially, this defines a homomorphic pairing function[1] for the integers. A Gaussian integer can be seen as the pair (a, b) where a and b are integers. This assigns a unique natural number to each one.

One can use this to show that pairs of integers have the same cardinality as natural numbers.

Further, since one can sum the binary representations of two numbers and get the representation of the result, this is homomorphic under addition. I have never before seen a pairing function with that property (not that I have seen many).

[1] https://en.wikipedia.org/wiki/Pairing_function#Cantor_pairin...