Complete lifts of harmonic maps and morphisms between Euclidean spaces

We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $\phi:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}$ (Proposition 2.3) and to construct many new examples of harmonic morphisms (Theorem 3.3). Finally we show that the complete lift of the quaternion product followed by the complex product is a simple and explicit example of a harmonic morphism which does not arise (see Definition 4.8 in \cite{BaiWoo95}) from any K{\"a}hler structure.