  We investigate the structure of a harmonic morphism $F$ from a Riemannian
4-manifold M^4 to a 2-surface $N^2$ near a critical point $m_0$. If $m_0$ is an
isolated 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 a
neighbourhood of $m_0$. If $M^4$ is compact without boundary, the singular
fibres of $F$ are branched minimal surfaces.
