Independent component analysis (ICA) is a class of algorithms widely applied to separate sources in EEG data. Most ICA approaches use optimization criteria derived from temporal statistical independence and are invariant with respect to the actual ordering of individual observations. We propose a method of mapping real signals into a complex vector space that takes into account the temporal order of signals and enforces certain mixing stationarity constraints. The resulting procedure, which we call Pairwise Complex Independent Component Analysis (PWC-ICA), performs the ICA in a complex setting and then reinterprets the results in the original observation space. We examine the performance of our candidate approach relative to several existing ICA algorithms for the blind source separation (BSS) problem on both real and simulated EEG data. On simulated data, PWC-ICA is often capable of achieving a better solution to the BSS problem than AMICA, Extended Infomax, or FastICA. On real data, the dipole interpretations of the BSS solutions discovered by PWC-ICA are physically plausible, are competitive with existing ICA approaches, and may represent sources undiscovered by other ICA methods. In conjunction with this paper, the authors have released a MATLAB toolbox that performs PWC-ICA on real, vector-valued signals.