A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.