The Gaussian integers usually aren't considered interesting enough to have disagreements about. They're in a weird spot because the integer restriction is almost contradictory with considering complex numbers: complex numbers are usually considered as how to express solutions to more types of polynomials, which is the opposite direction of excluding fractions from consideration. They're things that can solve (a restricted subset of) square-roots but not division.

This is really a disagreement about how to construct the complex numbers from more-fundamental objects. And the question is whether those constructions are equivalent. The author argues that two of those constructions are equivalent to each other, but others are not. A big crux of the issue, which is approachable to non-mathematicians, is whether it i and -i are fundamentally different, because arithmetically you can swap i with -i in all your equations and get the same result.