The goal of this paper is to present a new space, a complex valued controlled rectangular b-metric space (for short, υℂ-metric space). Some examples and topological properties of υℂ-metric spaces are given. Also, some related common fixed point results are discussed. Our results generalize a lot of works in this direction. Moreover, we apply the theoretical results to find a unique solution of a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. Finally, a numerical example to find the current that passes through the RLC circuit is illustrated.