  In 1980 Michelsohn defined a differential operator on sections of the complex
Clifford bundle over a compact K\"ahler manifold M . This operator is a
differential and its Laplacian agrees with the Laplacian of the Dolbeault
operator on forms through a natural identification of differential forms with
sections of the Clifford bundle. Relaxing the condition that M be K\"ahler, we
introduce two differential operators on sections of the complex Clifford bundle
over a compact almost Hermitian manifold which naturally generalize the one
introduced by Michelsohn. We show surprising K\"ahler-like symmetries of the
kernel of the Laplacians of these operators in the almost Hermitian and almost
K\"ahler settings, along with a correspondence of these operators to operators
on forms which are of present interest in almost complex geometry.
