Quaternionic regularity and the dibar-Neumann problem in C^2

Let Omega be a domain in the quaternionic space H. We prove a differential criterion that characterizes Fueter-regular quaternionic functions f : Omega -> H of class C1. We find differential operators T and N, with complex coefficients, such that a function f is regular on Omega if and only if (N-j T)f=0 on \partial\Omega ( j a basic quaternion) and f is harmonic on Omega. As a consequence, by means of the identification of H with C2, we obtain a non-tangential holomorphicity condition which generalizes a result of Aronov and Kytmanov. We also show how the differential criterion and regularity are related to the dibar-Neumann problem in C^2.