  For an algebraic number $\alpha$, the metric Mahler measure $m_1(\alpha)$ was
first studied by Dubickas and Smyth in 2001 and was later generalized to the
$t$-metric Mahler measure $m_t(\alpha)$ by the author in 2010. The definition
of $m_t(\alpha)$ involves taking an infimum over a certain collection
$N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of
Jankauskas and the author, the infimum in the definition of $m_t(\alpha)$ is
attained by rational points when $\alpha\in \mathbb Q$. As a consequence of our
main theorem in this article, we obtain an analog of this result when $\mathbb
Q$ is replaced with any imaginary quadratic number field of class number equal
to $1$. Further, we study examples of other number fields to which our methods
may be applied, and we establish various partial results in those cases.
