Let $M$ be a Riemannian 4-manifold. The associated twistor space is a bundle whose total space $Z$ admits a natural metric. The aim of this article is to study properties of complex structures on $Z$ which are compatible with the fibration and the metric. The results obtained enable us to translate some metric properties on $M$ (scalar flat, scalar-flat Kähler...) in terms of complex properties of its twistor space $Z$.