Quantum-like (QL) modeling, one of the outcomes of the quantum information revolution, extends quantum theory methods beyond physics to decision theory and cognitive psychology. While effective in explaining paradoxes in decision making and effects in cognitive psychology, such as conjunction, disjunction, order, and response replicability, it lacks a direct link to neural information processing in the brain. This study bridges neurophysiology, neuropsychology, and cognitive psychology, exploring how oscillatory neuronal networks give rise to QL behaviors. Inspired by the computational power of neuronal oscillations and quantum-inspired computation (QIC), we propose a quantum-theoretical framework for coupling of cognition/decision making and neural oscillations-QL oscillatory cognition. This is a step, may be very small, toward clarification of the relation between mind and matter and the nature of perception and cognition. We formulate four conjectures within QL oscillatory cognition and in principle they can be checked experimentally. But such experimental tests need further theoretical and experimental elaboration. One of the conjectures (Conjecture 4) is on resolution of the binding problem by exploring QL states entanglement generated by the oscillations in a few neuronal networks. Our findings suggest that fundamental cognitive processes align with quantum principles, implying that humanoid AI should process information using quantum-theoretic laws. Quantum-Like AI (QLAI) can be efficiently realized via oscillatory networks performing QIC.