  We define and study branched shadows of 4-manifolds as a combination of
branched spines of 3-manifolds and Turaev's shadows. We use these objects to
combinatorially represent 4-manifolds equipped with $Spin^c$-structures and
homotopy classes of almost complex structures. We then use branched shadows to
study complex 4-manifolds and prove that each almost complex structure on a
4-handlebody is homotopic to a complex one.
