As you know, the worst case for the Euclidean algorithm in $\mathbb Z$ is two consecutive Fibonacci numbers. As any online GCD calculator that shows the steps of the Euclidean algorithm will demonstrate, computing $\gcd(F_n, F_{n + 1})$ results in a listing in descending order of the Fibonacci numbers from $F_{n - 1}$ all the way down to 0 (assuming $n$ is positive to begin with, some of the online calculators refuse to do anything with negative numbers like $-89$ or $-144$).
What is the worst case for $\mathbb Z[i]$, the Euclidean domain of Gaussian integers?
The first thing I tried was $\gcd(F_n i, F_{n + 1})$, but it pretty much boils down to the same thing as in $\mathbb Z[i]$.
The second thing I tried was Googling. First result was Wikipedia. Ugh.
That's when I decided to ask the question here. I'm getting a warning that this question "appears subjective and is likely to be closed."
But I think there is an objective criterion here for what constitutes a bad case for the Euclidean algorithm.
The paper below attacks exactly this problem:
Heinrich Rolletschek, On the Number of Divisions of the Euclidean Algorithm Applied to Gaussian Integers, Journal of Symbolic Computation 2 (1986), no. 3, 261–291.
He proves an analogue of Lamé's theorem about the maximum number of steps but it seems that finding pairs that attain this maximum remains an open problem. (Or at least was in 1986.)