Novel computational systems different in principle from the von Neumann-type digital computers have long been investigated. As one of possible candidates for such novel computational systems, we outline a paradigm for a computation through a complex system in this paper. In particular, we address a complex computational system that utilizes high-dimensional analog physical chaotic dynamics suggested by the interactions between conscious and subconscious processes in the brain. Applications of the proposed system to combinatorial optimization are also given. In addition, we exploit many fixed points embedded in chaotic attractors with density of real-number, and fractal structures of the high-dimensional chaotic systems, in order to realize dynamical analog memories with a large memory capacity.