  In this paper we construct complex contact structures on $\mathbb{C}^{2n+1}$
for any $n\ge 1$ with the property that every holomorphic Legendrian map
$\mathbb{C}\to \mathbb{C}^{2n+1}$ is constant. In particular, these contact
structures are not globally contactomorphic to the standard complex contact
structure on $\mathbb{C}^{2n+1}$.
