Most of the time we subconsciously determine the ring a GCD computation is to take place in purely from context. For example, $\gcd(28, 63)$ is clearly a GCD computation in $\textbf Z$, while $\gcd(28, 9 + 3 \sqrt 2)$ takes place in $\textbf Z[\sqrt 2]$. I think the answer of the former pair would be the same in the latter ring, right?
What about something like $\gcd(21 - 21 \sqrt 3, 9 + 3 \sqrt 2)$? It seems to me the answer is neither in $\textbf Z[\sqrt 2]$ nor $\textbf Z[\sqrt 3]$. I'm guessing the GCD would be in the ring of integers of $\textbf Q(\sqrt 2 + \sqrt 3)$, which is of degree $4$.
In general, if the GCD can't be resolved in $\textbf Z$ nor in either of the rings of least algebraic degree containing the two operands, would the ring of least algebraic degree containing the GCD be of algebraic degree equal to the product of the algebraic degrees of the two operands?