This chapter introduces a novel two-dimensional fractional discrete memristor-based map that exhibits rich and complex dynamics, including hidden attractors, multistability, and hyperchaotic behavior. By embedding a Caputo-type fractional-order memory into a discrete non-equilibrium map structure, the model captures essential nonlinear and memory-driven features that conventional integer-order systems fail to describe. The system lacks fixed points, ensuring the existence of hidden attractors. A combination of analytical techniques and numerical simulations—such as Lyapunov exponent computation, bifurcation analysis, phase portraits, and complexity measures (0–1 test and \(C_0\) entropy)—is used to explore the system’s behavior over a wide parameter space. The study demonstrates how fractional memory alters bifurcation routes and amplifies dynamical richness, establishing the proposed model as a valuable framework for chaos-based computation, secure communication, and intelligent system design.