A bidirectional associative memory (BAM) with complex states and connection weights is investigated in this paper. The states are represented by quantization values defined on the unit circle of the complex plane. A given Lyapunov function indicates that the proposed complex domain BAM (CDBAM) is bidirectionally stable no matter which operation (synchronous or asynchronous) is used. We also prove that all the equilibrium (fixed) points of CDBAM correspond to local energy minima so that the design problem can be solved by a gradient descent algorithm. Finally, a gradient descent algorithm in the complex domain is derived to design the weight matrix of CDBAM. Several computer simulations are performed to illustrate the validity, capacity, attractivity and the applications of the CDBAM.