To extend the existing real-valued BCS (Bayesian compressive sensing) framework to the complex-valued one, a CMBCS-MLP (complex multitask Bayesian compressive sensing algorithm using modified Laplace priors) was developed to eliminate the impact of measurement noise variance, and a fast algorithm based on sequential operations was further derived. It is demonstrated by numerical examples that the developed CMBCS-MLP algorithm is more accurate and robust than the existing algorithms in the complex sparse signal reconstructions.