The exploration of complex-valued chaos not only provides a feasible approach for practical applications such as image encryption, but also has great potential in simulating wave phenomena and quantum inspired process. In order to bridge it with nonlinear circuit components, we introduce a novel complex-valued chaotic system by embedding a discrete memristor into a complex Gaussian map. The memristor, a component with inherent physical memory, is uniquely driven by the modulus of the complex state variable, which is a key physical quantity often related to energy or amplitude in wave systems. This coupling induces complex nonlinear dynamics, which are physically characterized through Lyapunov exponents and bifurcation analysis, revealing an enhanced and more robust chaotic regime. The physical feasibility of this system is demonstrated by its successful hardware realization on an FPGA platform. To demonstrate its potential applications, we leverage the complex chaotic flows of the system to engineer a dual-image encryption scheme, where the encryption process is explained as a physical diffusion and scrambling of information represented by a complex matrix. Our results verify that this approach not only yields a cryptosystem with high security but also provides a link between complex chaos and information security applications.