  We study complex Dirac structures, that is, Dirac structures in the
complexified generalized tangent bundle. These include presymplectic
foliations, transverse holomorphic structures, CR-related geometries and
generalized complex structures. We introduce two invariants, the order and the
(normalized) type. We show that, together with the real index, they allow us to
obtain a pointwise classification of complex Dirac structures. For constant
order, we prove the existence of an underlying real Dirac structure, which
generalizes the Poisson structure associated to a generalized complex
structure. For constant real index and order, we prove a splitting theorem,
which gives a local description in terms of a presymplectic leaf and a small
transversal.
