Traditional real-valued neural networks often suffer from high computational costs, limited robustness, and difficulty capturing phase information, which is critical for understanding wave and rotation-related phenomena such as electromagnetism, light waves, quantum waves, and oscillatory phenomena. Nature often defies real-number descriptions, as seen in equations like x^2 = -1, which has no solution in the real domain but is naturally resolved within the complex domain. Complex-valued neural networks (CVNNs) offer a promising solution, enabling a richer and more accurate representation of real-world problems, such as MRI, speech processing, and biological signal analysis. This research aims to develop both theoretical and practical foundations for computationally efficient components of complex-valued convolutional neural networks (CVNNs), grounded in complex number algebra and signal processing theory. The study will focus on the design challenges of complex-valued activation functions, considering the Cauchy-Riemann equations (ensuring complex differentiability) and Liouville’s theorem (which states that a bounded holomorphic function must be constant). These theoretical principles will guide the development of new non-linear activation functions that balance expressiveness with mathematical robustness. Building on these findings, the research will design computationally efficient network components and optimise training strategies to enhance the performance and robustness of CVNNs for real-world tasks. Additionally, the study will investigate the resilience of CVNNs to adversarial attacks, aiming to improve their robustness in practical applications. Moreover, this study will investigate the integration of Quantum Neural Networks (QNNs), recognising that the mathematical structure of CVNNs shares strong parallels with quantum models. Both CVNNs and QNNs leverage complex-number computations to capture phase and higher-order dependencies. While CVNNs are not inherently quantum, they can inspire quantum-aware architectures that align more closely with the principles of quantum information processing. This direction has the potential to bridge conventional AI and quantum computing, unlocking more accurate and efficient models for future AI systems. Ultimately, this research seeks to advance AI by aligning machine learning models more closely with natural processes, unlocking new opportunities for both theoretical development and practical applications.