The functioning of complex natural systems, such as living systems, still lacks a generally accepted theoretical basis and respective empirical verification. We propose an experiment to test whether such systems are subject to a hitherto unknown organizing principle, in which the quantum mechanical uncertainty principle enables ordering phenomena in chaotic or nearly chaotic systems in the sense of a strong emergence principle. Contrary to previous proposals, our hypothesis does not require fragile macroscopic quantum coherence, ensuring compatibility with the harsh conditions in living systems. To test this hypothesis, we suggest comparing two virtually identical, complex artificial neural networks as a quantitative benchmark. One setup utilizes deterministic pseudo-random number generators at key sensitive points, while the other uses quantum-based physical random-number generators. A favorable outcome demonstrating the superiority of quantum randomness would have considerable implications beyond artificial networks, particularly for the discussions on consciousness and free will.