  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.
