Abstract. This is a report on joint work concerning almost complex structures on a quaternion K\" ahler manifold \mqk which are compatible with the quaternionic structure $Q$, see \cite{AM1,2}, \cite{P1,2} and \cite{AMP1,2,3}. We point out the interest for conformally balanced complex structures, which are related to solutions of a ``twistor equation" and on a compact positive \qk manifold may be seen as Hamiltonian forms of Killing vector fields on $M^{4n}$ or as Hamiltonian functions on the twistor space $Z$ and also as eigenforms of the Laplacian acting on self-dual 2-forms.
AMSclassification. 53C10, 53C15, 32C10, 58G25
Keywords. Quaternion K\"ahler - Compatible complex structure - Conformally balanced complex structure - Hamiltonian form - Self-dual form