Hermitian structures and harmonic morphisms in higher dimensional Euclidean spaces

We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from ${\mathbf{R}}^n$ for each $n>4$ which do not arise from a K"ahler structure; it is known that such examples do not exist for $n\le 4$.