The paper contains description of the orthogonal complex structures with respect to the natural 1-parameter family of Riemannian metrics on the (negative) twistor space over a self-dual Einstein Riemannian 4-manifold. We prove that if the twistor space of a compact self-dual Einstein 4-manifold admits more than one orthogonal complex structure then the 4-manifold has a Kähler structure. Considering the flag manifold F _1,2 which is the twistor space of CP ² endowed with the Fubini–Study metric, we obtain that any invariant Einstein metric on F _1,2 admits even locally exactly three orthogonal complex structures which are the invariant ones.