Inspired by the all-important conformal invariance of harmonic maps on
two-dimensional domains, this article studies the relationship between biharmonicity
and conformality. We first give a characterization of biharmonic morphisms,
analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in
particular, explicits the conditions required for a conformal map in dimension four
to preserve biharmonicity and helps producing the first example of a biharmonic
morphism which is not a special type of harmonic morphism. Then, we compute the
bitension field of horizontally weakly conformal maps, which include conformal
mappings. This leads to several examples of proper (i.e., non-harmonic) biharmonic
conformal maps, in which dimension four plays a pivotal role. We also construct a
family of Riemannian submersions which are proper biharmonic maps.