Abstract. We study maps between complex {\mfd}s which preserve, by post-composition, local {\ph} functions. We characterise such maps, called {\em {\ph} {\mo}s}, as {\pho} maps.
AMSclassification. Primary 32F05; Secondary 31C10, 53C55, 58E20
Keywords. Pluriharmonic maps, harmonic maps, pluriharmonic morphisms