Quantum duality, or wave-particle duality, is not directly dependent on the mathematical description of an object. It is a fundamental property of quantum systems, irrespective of the mathematical formalism used to describe them. Wave-particle duality refers to the observation that particles can exhibit both wave-like and particle-like behavior.
The mathematical framework used to describe quantum systems, such as wavefunctions in quantum mechanics or state vectors in quantum field theory, provides a formal representation of the system's behavior and allows for predictions of measurement outcomes. These mathematical descriptions involve complex numbers, not just real numbers, to account for the interference and superposition of probabilities that are characteristic of quantum phenomena.
However, it is worth noting that when we measure or observe a quantum system, the outcomes of measurements are typically associated with real numbers. In experiments, we often measure quantities such as position, momentum, energy, or spin, which are represented by real values. The probabilistic nature of quantum mechanics allows us to calculate the probabilities of obtaining different measurement outcomes, but the actual measurement result is a real number.
In summary, the wave-particle duality of quantum systems is a fundamental aspect of nature that is not dependent on the mathematical formalism used to describe it. While complex numbers are used in the mathematical framework of quantum mechanics, the measurement outcomes themselves are typically associated with real numbers.