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?

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.