  In this paper we introduce the notion of a formal complex contact structure
on an odd dimensional complex manifold. Our main result is that every formal
complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic
contact structure on a Stein domain $\Omega\subset X$ which is diffeotopic to
$X$. We also prove a parametric h-principle in this setting, analogous to
Gromov's h-principle for contact structures on smooth open manifolds. On Stein
threefolds we obtain a complete homotopy classification of formal complex
contact structures. Our methods also furnish a parametric h-principle for germs
of holomorphic contact structures along totally real submanifolds of class
$\mathscr C^2$ in arbitrary complex manifolds.
