  In this work, we prove that every complex contact structure gives rise to a
distinguished type of almost contact metric 3-structure. As an application of
our main result, we provide several new examples of manifolds which admit taut
contact circles, taut and round almost cosymplectic 2-spheres, and almost
hypercontact (metric) structures. These examples generalize, in a suitable
sense, the well-known examples of contact circles defined by the
Liouville-Cartan forms on the unit cotangent bundle of Riemann surfaces.
Furthermore, we provide sufficient conditions for a compact complex contact
manifold to be the twistor space of a positive quaternionic K\"{a}hler
manifold. In the particular setting of Fano contact manifolds, from our main
result, we also obtain new evidences supporting the LeBrun-Salamon conjecture.
