  The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the
two-dimensional (2D) global conformal group on the celestial sphere at infinity
where asymptotic 4D scattering states are specified. Consequent similarities of
4D flat space amplitudes and 2D correlators on the conformal sphere are
obscured by the fact that the former are usually expressed in terms of
asymptotic wavefunctions which transform simply under spacetime translations
rather than the Lorentz $SL(2,\mathbb{C})$. In this paper we construct on-shell
massive scalar wavefunctions in 4D Minkowski space that transform as
$SL(2,\mathbb{C})$ conformal primaries. Scattering amplitudes of these
wavefunctions are $SL(2,\mathbb{C})$ covariant by construction. For certain
mass relations, we show explicitly that their three-point amplitude reduces to
the known unique form of a 2D CFT primary three-point function and compute the
coefficient. The computation proceeds naturally via Witten-like diagrams on a
hyperbolic slicing of Minkowski space and has a holographic flavor.
