  There is a classical extension, of M\"obius automorphisms of the Riemann
sphere into isometries of the hyperbolic space $\mathbb{H}^3$, which is called
the Poincar\'e extension. In this paper, we construct extensions of rational
maps on the Riemann sphere over endomorphisms of $\mathbb{H}^3$ exploiting the
fact that any holomorphic covering between Riemann surfaces is M\"obius for a
suitable choice of coordinates.
  We show that these extensions define conformally natural homomorphisms on
suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex
multiplication to a product in $\mathbb{H}^3$ that allows to construct a visual
extension of any given rational map.
