  Let D be a domain in the quaternionic space H. We prove a differential
criterion that characterizes Fueter-regular quaternionic functions f:bD -> H of
class C^1. We find differential operators T and N, with complex coefficients,
such that a function f is regular on D if and only if (N-jT)f=0 on the boundary
of D (j a basic quaternion) and f is harmonic on D. As a consequence, by means
of the identification of H with C^2, 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.
