Much progress has been made in the research of synchronization for chaotic real or complex nonlinear systems. In this paper we introduce a new type of synchronization which can be studied only for chaotic complex nonlinear systems. This type of synchronization may be called complex lag synchronization (CLS). A definition of CLS is introduced and investigated for two identical chaotic complex nonlinear systems. Based on Lyapunov function a scheme is designed to achieve CLS of chaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modulus errors and phase errors of these chaotic attractors after synchronization to prove that CLS is achieved.