As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful Kahler identities.

Can I ask what extra structure we get on $\Omega(M)$ if we also assume the existence of a quaternion-Kähler structure? I assume here that $M$ has non-zero Ricci curvature, since in the flat case it is well-known that we get a representation of the quaternions of $\Omega(M)$, which is to say, we get a hyper-Kahler manifold.