In literature, Fixed point theorems have been proved for a single-valued self-mapping using the constant of contraction conditions in the classical metric spaces. Many findings have been obtained by researchers on fixed points of various classes of mappings defined on generalized metric spaces. In this work, on bi-complex valued metric spaces, we have proved common fixed point results for a pair of self-mappings using rational type contraction conditions involving three-variable control functions in which one variable is fixed. Thus we have extended and improved the contraction conditions of many existing theorems from the constant of contraction to three variable control functions on bi-complex valued metric spaces. Also, we have narrowed down and improved the contraction conditions from the whole space to the closed ball. As a consequence, by using the appropriate locally point-dependent control functions with constant coefficients and mappings in our main theorems, we have derived many results that exist in the literature on a closed ball in bi-complex valued complete metric spaces. Added to that, we have given some examples to verify our results and as an application, we have solved the system of Urysohn integral equations using our finding.