TITLE: In what quadrant should the GCD of two Gaussian integers with both real and imaginary parts be?
QUESTION [7 upvotes]: Examples:
$$\gcd(-9 - 3i, -2 + i)$$
$$\gcd(9 + 3i, -2 - i)$$
$$\gcd(-9 - 3i, -2 - i)$$
$$\gcd(-3 - 9i, -2 + i)$$
$$\gcd(-3 + 9i, -1 + 2i)$$
I've put these through Wolfram Alpha and, for some, gotten exactly the result I expected, but for some I have been surprised by the results, but I'm well aware that WA can in some cases get confused and arrive at the wrong result.
For justification of the answers, however nonsensical they may seem, I'm more interested in something that follows logically and by commonsense from analogy to $\mathbb{Z}$ and the Euclidean algorithm, rather than something that involves some exotic mathematical object presented as part of an appendage measuring contest.
Bonus question: how neatly does this carry over to other quadratic imaginary rings that are Euclidean? What about reals?

REPLY [3 votes]: Every gaussian prime $z$ has four associates
$$z, iz,  -z,  -iz$$
There will always be exactly one of those associates in the first quadrant or on the positive real axis.
If you want some sort of uniqueness to prime factorization, then you could do it this way.