In our previous work, we initiated the study of dynamics in networks with identical nodes that evolve under complex quadratic iterations. For these Complex Quadratic Networks, (or CQNs for short), we focused on defining the equi-M set as the natural extension of the Mandelbrot set, representing the locus of complex parameters for which the network's critical orbit remains bounded. This paper examines the structure and limitations of equi-M sets in two-dimensional Complex Quadratic Networks (CQNs). In particular, we aim to describe the relationship between the equi-M set and the parameter domains where the critical orbit converges to periodic attractors. The simplest case of two coupled nodes serves as a foundational testbed: its analytical tractability enables the identification of critical phenomena and their dependence on coupling, while offering insight into more general principles. The two-node case is also simple enough to allow for explicit characterization of the coupling conditions that govern transitions between synchronized and desynchronized behavior. Using a combination of analytical and numerical methods, the study reveals that while the parameter region corresponding to an attracting fixed point closely tracks the boundary of the equi-M set near its main cusp, this correspondence breaks down for higher periods and in regions supporting coexisting attractors. These discrepancies highlight key differences between uncoupled one-dimensional dynamics and coupled systems, where equi-M sets capture only partial information about the system's global combinatorics. Overall, the results illustrate how coupling transforms the familiar structure of the Mandelbrot set into a richer, higher-dimensional landscape for low-dimensional CQNs and point toward a sharp increase in complexity as the number of nodes grows. This work bridges classical complex dynamics with emerging questions in networked nonlinear systems and lays the groundwork for analyzing high-dimensional coupled dynamics.