Abstract
In this paper, with the use of generalized complex and hyperbolic numbers, we build the theory of generalized quaternions with hyperbolic-generalized complex ( \(\mathcal{H}\mathcal{G}\mathcal{C}\) ) numbers as coefficients. Additionally, certain associated theoretical universal results involving \(\mathcal{H}\mathcal{G}\mathcal{C}\) Fibonacci and Lucas numbers, including their generalized quaternions, are established. With this approach, bihyperbolic, hyperbolic-complex, and hyperbolic-dual generalized quaternions can be determined for specified values of \(\mathfrak{p} \in \mathbb{R}\) . It is also possible to study numerous types of quaternions with \(\mathcal{H}\mathcal{G}\mathcal{C}\) number coefficients and their attributes depending on the choice of the real values \(\alpha \) and \(\beta \) .
