<abstract><p>Complex interval-valued intuitionistic fuzzy sets not only consider uncertainty and periodicity semantics at the same time but also choose to express the information value with an interval value to give experts more freedom and make the solution to the problem more reasonable. In this study, we used the interval quaternion number space to generalize and extend the utility of complex interval-valued intuitionistic fuzzy sets, analyze their order relation, and offer new operations based on interval quaternion numbers. We proposed a new score function and correlation coefficient under interval quaternion representation. We applied the interval quaternion representation and correlation coefficient to a multi-criterion decision making model and applied the model to enterprise decision-making.</p></abstract>