  We classify complex surfaces $(M,\,J)$ admitting Engel structures
$\mathcal{D}$ which are complex line bundles. Namely we prove that this happens
if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of
such Engel structures by adapting a construction due to Geiges. We also study
associated Engel defining forms and define a unique splitting of $TM$
associated with $\mathcal{D}$ $J$-Engel.
