Proposed Test for Complex versus Quaternion Quantum Theory via Quantum Superdense Coding

While complex numbers form the foundational number system of quantum mechanics, the theoretical possibility of higher-dimensional hypercomplex alternatives persists. In this paper, we present a test protocol designed to distinguish between complex and quaternionic quantum mechanics. The protocol is founded upon the quantum superdense coding scheme and functions by detecting the presence of multiple distinct quantum phase gates. We also provide the corresponding quantum circuit required to perform this test.