  Let $L/k$ be a finite abelian extension of an imaginary quadratic number
field $k$. Let $\mathfrak{p}$ denote a prime ideal of $\mathcal{O}_k$ lying
over the rational prime $p$. We assume that $\mathfrak{p}$ splits completely in
$L/k$ and that $p$ does not divide the class number of $k$. If $p$ is split in
$k/\mathbb{Q}$ the first named author has adapted a construction of Solomon to
obtain elliptic $\mathfrak{p}$-units in $L$. In this paper we generalize this
construction to the non-split case and obtain in this way a pair of elliptic
$\mathfrak{p}$-units depending on a choice of generators of a certain Iwasawa
algebra (which here is of rank 2). In our main result we express the
$\mathfrak{p}$-adic valuations of these $\mathfrak{p}$-units in terms of the
$p$-adic logarithm of an explicit elliptic unit. The crucial input for the
proof of our main result is the computation of the constant term of a suitable
Coleman power series, where we rely on recent work of T. Seiriki.
