  We establish a correspondence between a conformally invariant complex scalar
field action (with a conformal self-interaction potential) and the action of a
phantom scalar field minimally coupled to gravity (with a cosmological
constant). In this correspondence, the module of the complex scalar field is
used to relate conformally the metrics of both systems while its phase is
identified with the phantom scalar field. At the level of the equations, the
correspondence allows to map solution of the conformally non-linear
Klein-Gordon equation with vanishing energy-momentum tensor to solution of a
phantom scalar field minimally coupled to gravity with cosmological constant
satisfying a massless Klein-Gordon equation. The converse is also valid with
the advantage that it offers more possibilities owing to the freedom of
rewriting a metric as the conformal transformation of another metric. Finally,
we provide some examples of this correspondence.
