  Let Z be a compact complex (2n+1)-manifold which carries a {\em complex
contact structure}, meaning a codimension-1 holomorphic sub-bundle D of TZ
which is maximally non-integrable. If Z admits a K\"ahler-Einstein metric of
positive scalar curvature, we show that it is the Salamon twistor space of a
quaternion-K\"ahler manifold (M^{4n}, g). If Z also admits a second complex
contact structure, then Z= CP_{2n+1}. As an application, we give several new
characterizations of the Riemannian manifold HP_n=Sp(n+1)/( Sp(n)\times Sp(1)).
