Two-dimensional IIR filters are designed using an error criterion consisting of the weighted sum of magnitude, phase, and stability errors, thereby extending the spectral factorization-based method to complex approximation. The weights are adjusted to effectively constrain stability while minimizing a weighted sum of frequency domain magnitude and phase errors. We use a derivative-free variant of the Marquardt optimization algorithm augmented by a one-dimensional search. We present examples of approximation of both linear and nonlinear phase ideal transfer functions, and conduct comparisons across a range of numerator and denominator orders, from FIR to all-pole, while keeping the total number of coefficients approximately constant. We find that the designed IIR filters offer better magnitude response in the case of linear phase ideal functions, even considering coefficient reflection symmetry, but of course, the FIR filter has the better (zero) phase error. When the ideal transfer function has nonlinear phase, and hence no coefficient reflection symmetry, we find that the designed IIR filter can perform better than the FIR filter with regard to both magnitude and phase error, when magnitude error is weighted more heavily than phase error.