The paper introduces a novel memristive chaotic system characterized by an infinite number of index-2 saddle foci, enabling it to generate multiscroll chaos and exhibit extreme multistability. Bifurcation diagrams, phase portraits and other methods are employed to examine the stabilities of equilibria and complex dynamics. It shows that by modifying the function of memristor, the system can produce multiscroll attractors with varying scroll counts. Furthermore, it can be decomposed into coexisting chaotic attractors at different locations, and this decomposition is influenced by adjustments in parameters and initial values, illustrating the impact of initial-relied and parameter-relied offset boosting. With variations in the parameter, the coexisting chaotic attractors will undergo a bifurcation, ultimately transforming into coexisting periodic attractors. The image encryption application of the system is explored, introducing an efficient chaos-based algorithm applied to encrypt Internet of Medical Things (IoMT) images, followed by a comprehensive performance evaluation. •New memristive system with multiscroll chaos and offset boosting is presented.•Complex dynamics of the system are theoretically and numerically studied.•New chaos-based image encryption algorithm with high security is designed.