Compatible complex structures on almost quaternionic manifolds

On an almost quaternionic manifold (M-4n, Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n greater than or equal to 2, we prove that the existence of two compatible complex structures I-1, I-2 not equal +/-I-1 forces (M-4n, Q) to be quaternionic. If n = 1, that is (M-4, Q) = (M-4, [g], or) is an oriented conformal 4-manifold, we prove a maximum principle for the angle function [I-1, I-2] of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twister space Z of an almost quaternionic manifold (M-4n, Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twister constructions.