Independent component analysis (ICA) for separating complex-valued sources is needed for convolutive source-separation in the frequency domain, or for performing source separation on complex-valued data, such as functional magnetic resonance imaging data. Previous complex infomax approaches have proposed using bounded (and hence non-analytic) nonlinearities. We propose using an analytic (and hence unbounded) complex nonlinearity for infomax for processing complex-valued sources. We show that using an analytic nonlinearity for processing complex data has a number of advantages. First, when compared to split-complex approaches (i.e., approaches that split the real and imaginary data into separate channels), the shape of the performance surface is improved resulting in better convergence characteristics. Additionally, the computational complexity is significantly reduced, and finally, the presence of cross terms in the Jacobian enables the analytic nonlinearity to approximate a more general class of input distributions.