Representation of Quaternionic $\psi$-hyperholomorphic Functions

In the algebra of complex quaternions $\mathbb{H}(\mathbb{C})$ we consider for the first time left- and right-$\psi$-hyperholomorphic functions. We justify the transition in left- and right-$\psi$-hyperholomorphic functions to a simpler basis i.e. to the Cartan basis. Using Cartan’s basis we find the solution of Cauchy-Fueter equation. By the same method we find a representation of left-$\psi$-hyperholomorphic function in a special case.