[math-fun] complex continued fractions

Continued fractions of arbitrary complex numbers can be computed by use of a Gaussian gcd-type recursion. There's lots of info about the cf expansions of pi, e, sqrt(2), phi, etc. Is there anything known about the cf expansions of e+%i*pi, sqrt(2)+%i*sqrt(3), etc.? If anything interesting about such expansions were found, would that say anthing interesting about the real or imaginary parts? A lot of the fun cf expansions of various elementary functions still "work" when the arguments aren't too far off the real axis.