Let \((M,g)\) be an oriented self-dual compact Einstein 4-manifold \((M,g)\) and \(Z\) its twistor space with the standard Riemannian metric \(h_t\) which depends on the parameter \(t>0\). It is well-known that \(Z\) admits a canonical complex structure \(J_Z\) which is orthogonal with respect to any standard metric \(h_t\), \(t>0\). The authors prove that if \(Z\) admits (for any \(t>0\)) a locally defined \(h_t\)-orthogonal positively oriented complex structure, different from \(\pm J_Z\), then \((M,g)\) is one of the following compact Riemannian 4-manifolds: \(\mathbb C P^2\) with the Fubini-Study metric, a flat compact 4-manifold, a K3 surface or its quotients by the groups \(\mathbb Z_2\) or \(\mathbb Z_2 \times \mathbb Z_2\) with a Calabi-Yau Ricci flat metric, or a compact quotient of the complex hyperbolic plane with the standard metric. Moreover, in the case \(M=\mathbb C P^2\) and, hence, \(Z = SU(3)/T^2\), any \(h_t\)-orthogonal complex structure on \(Z\) is \(SU(3)\)-invariant and it coincides (up to a sign) with one of three commuting invariant complex structures.