For complex numbers in general, you can just divide a/b. I'm assuming you're referring to complex numbers where the real and imaginary parts are integers, those are called Gaussian Integers and you can indeed use a kind of euclidean algorithm. Any system with this property is called an Euclidean domain, see this link for an example using the Gaussian Integers http://mathforum.org/library/drmath/view/67068.html
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