  We propose and study a generalized continued fraction algorithm that can be
executed in an arbitrary imaginary quadratic field, the novelty being a
non-restriction to the five Euclidean cases. Many hallmark properties of
classical continued fractions are shown to be retained, including exponential
convergence, best-of-the-second-kind approximation quality (up to a constant),
periodicity of quadratic irrational expansions, and polynomial time complexity.
