  We obtain all possible parameters of Plotkin-optimal two-Lee weight
projective codes over $\mathbb{Z}_4,$ together with their weight distributions.
We show the existence of codes with these parameters as well as their weight
distributions by constructing an infinite family of two-weight codes.
Previously known codes constructed by Shi et al. (\emph{Des Codes Cryptogr.}
{\bf 88}(3):1-13, 2020) can be derived as a special case of our results. We
also prove that the Gray image of any Plotkin-optimal two-Lee weight projective
codes over $\mathbb{Z}_4$ has the same parameters and weight distribution as
some two-weight binary projective codes of type SU1 in the sense of Calderbank
and Kantor (\emph{Bull. Lond. Math. Soc.} {\bf 18}:97-122, 1986).
