Observation data obtained from the Four-Dimensional (4D) Synthetic Aperture Radar (SAR) system is sparse and non-uniform in the baseline-time plane. Hence, the imaging results acquired by traditional Fourier-based methods are limited by high side lobes. Compressive Sensing (CS) is a recently proposed technique that allows for the recovery of an unknown sparse signal with overwhelming probability from very limited samples. However, the standard CS framework has been developed for real-valued signals, and the imaging process for 4D synthetic aperture radar deals with complex-valued data. In this study, we propose a new 4D synthetic aperture radar imaging algorithm based on an iterative reconstruction of magnitude and phase, which transforms the height-velocity imaging problem of 4D synthetic aperture radar into a joint reconstruction problem of the magnitude and phase of the complex-valued scatter coefficient. Using the phase information in the algorithm, the image quality is improved. Simulation results confirm the effectiveness of the proposed method.