Continued fractions for complex numbers can be studied. However, there is no "canonical" choice of which possible continued fraction to use. And, for that matter, why use the Gaussian integers $\mathbb Z[\sqrt{-1}\;]$ and not some other chioce?
See HERE for an article:
Convergence of Complex Continued Fractions
by John Marafino (James Madison University) and Timothy J. McDevitt (James Madison University)
This article originally appeared in:
Mathematics Magazine
June, 1995
Convergence of a complex continued fraction can be analyzed using analysis, algebra, number theory, topology or complex analysis.