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We investigate the structure of a harmonic morphism $F$ from a Riemannian4-manifold M^4 to a 2-surface $N^2$ near a critical point $m_0$. If $m_0$ is anisolated critical point or if $M^4$ is compact without boundary, we show that$F$ is pseudo-holomorphic w.r.t. an almost Hermitian structure defined in aneighbourhood of $m_0$. If $M^4$ is compact without boundary, the singularfibres of $F$ are branched minimal surfaces.

On harmonic morphisms from 4-manifolds to Riemann surfaces and local almost Hermitian structures