This chapter explores the intricate dynamics and complexity of the fractional 4D Grassi-Miller memristive-based map, a novel extension of discrete nonlinear systems. By incorporating fractional-order calculus, the map exhibits a wide range of dynamical behaviours, including hidden attractors, chaotic oscillations, and hyperchaotic states, even in the absence of fixed points. The memristive G-M map’s trajectories are systematically analyzed using a suite of advanced nonlinear tools, including phase portraits, bifurcation diagrams, Lyapunov exponents, the 0–1 test for chaos, and permutation entropy. These techniques collectively validate the presence of high-dimensional chaos and complex attractor structures, including hidden dynamics not associated with equilibria. The combined use of entropy measures and chaos indicators enables the detailed mapping of transitions between periodic, quasiperiodic, and hyperchaotic states across the parameter space. Numerical simulations further support these findings, highlighting the critical influence of fractional memory and parameter sensitivity on the emergence of complex behaviors. This comprehensive study establishes the Grassi-Miller map as a powerful paradigm for modeling memory-driven nonlinear phenomena.