  In the algebra of complex quaternions $\mathbb{H(C)}$ 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.
