Complex number arithmetic computations are one of the key arithmetic components in modern digital communication and optical systems. Complex number multiplication and complex number inner-product play a unique role in these applications. In this paper, a complex-number multiplier and complex-number inner-product processor based on a Redundant Binary (RB) representation are presented. This work is an extension of a previous real fixed-point inner-product hardware design. With the proposed algorithms, the complex number multiplication is reduced to parallel RB multiplications, and the complex number inner-product is produced using a RB addition tree. This proposed inner-product processor can be reconfigured or controlled to perform different computations such as inner-product processing or parallel multiplies for real and/or complex numbers. The design results, not only in simplified arithmetic operations, but also in a highly parallel and simple architecture when compared with other methods.