Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, while virtually all fMRI studies only use the magnitude of the data in the analysis. Since little is known for devising models for the phase, independent component analysis (ICA) emerges as a promising technique for data-driven analysis of fMRI data in its native complex form. In this paper, we compare the performance of ICA on real-valued and complex-valued fMRI data and show the advantages of the complex approach. We also develop complex-valued order selection scheme to improve the estimation of the number of independent components in complex-valued fMRI data using information-theoretic criteria. Comparisons on order selection using real-valued and complex-valued fMRI data demonstrate the more informative nature of complex data.