In optical coherence tomography (OCT), images are usually formed from the envelope of the measured interference signal. Computation of the absolute magnitude of the signal for measurement of the envelope is a nonlinear process that destroys phase information. This study explores the idea of recording and processing the phase of the OCT interference signal before calculation of the magnitudes for display. Processing the partially coherent OCT signals in the complex domain provides the opportunity to correct phase aberrations responsible for speckle noise in OCT images. We describe an OCT system that incorporates a quadraturedemodulation scheme for accurate recording of the phase and amplitude of OCT signals from single or multiple detectors. A speckle-reduction technique that works in the complex domain, called the zeroadjustment procedure (ZAP), is investigated as an example of complex-domain processing. After demonstrating its speckle-correction properties mathematically and in numerical simulations, we apply ZAP to OCT images of living skin. The results show that ZAP reduces speckle contrast in regions where scatterer density is high and expands the range of gray values in the image. However, as presently implemented, ZAP tends to blur sharp boundaries between image features.