An Axiomatic Theory of Quaternionic Probability Extending Kolmogorov’s Framework

We present an axiomatic construction of quaternionic probability, extending Kolmogorov’s classical framework to the noncommutative algebra of quaternions. The theory introduces quaternionic probability spaces, conditional probabilities, Bayes’ rules, independence, random variables, expectations, and transport equations, all formulated in a consistent manner. Classical probability is recovered through scalar projection, while restriction to complex subalgebras reproduces the standard quantum formalism. Uniquely quaternionic structures arise, including noncommutative conditional probabilities, inequivalent forms of independence, and quaternionic transport laws. The framework further develops quaternionic Markov chains, entropy, and divergence measures that separate scalar uncertainty from vectorial coherence. Several illustrative examples are provided to show how quaternionic probability captures order effects, hidden correlations, and orthogonal divergences—features invisible to both classical and complex approaches. These results establish quaternionic probability as a rigorous generalization of Kolmogorov’s axioms and as a potential foundation for future studies in noncommutative probability, integrable structures, and quaternionic extensions of mathematical physics.