Throughout this paper, a new generalization of complex valued CR  -metric space is used. This definition was stated in published research as a previous work. In this paper, some new results on common best proximity points for non-self-mappings are obtained in these spaces. It is an extension of previous results proven in classical complex metric spaces. The requirements of the work are as follows: first, we present some equivalent statements for convergence and Cauchy sequences. Second, we define complex b- rectangular metric space by using the function distance of a recent space. Finally, we present the concepts of commute proximity, weak - property and weakly dominate mappings. These concepts are illustrated by some examples. Furthermore, some new results on common best proximity points for non-self-mappings are obtained in these spaces.  As well as the extension of previous results have been proven in classical complex metric spaces. The work required the employment of conditions on mappings, which are commute proximity and continuity and one of the mappings weakly dominates other. In addition to other appropriate conditions. As a direct result, the existence of a common unique fixed point has been proved.