  Given a compact complex manifold, we develop Hodge theory for the elliptic
complex of differential forms defined by Bigolin in 1969 and recently referred
as the Schweitzer complex. We exhibits several $L^2$ orthogonal decompositions
of spaces of forms and prove a Hodge decomposition for harmonic forms on
compact K\"ahler manifolds. Then we compute the cohomology of this complex on
the small deformations of the complex structure of the Iwasawa manifold,
showing that this cohomology is as powerful as Aeppli and Bott-Chern
cohomology, in order to distinguish classes of complex structures. Finally, we
partially extend the definition of this complex on almost complex manifolds,
providing a new cohomological invariant on 1-forms which is finite dimensional
when the manifold is compact.
