  We describe completely conformal Killing or conformal Killing-Yano (CKY)
$p$-forms on almost abelian metric Lie algebras. In particular we prove that if
a $n$-dimensional almost abelian metric Lie algebra admits a non-parallel CKY
$p$-form, then $p=1$ or $p=n-1$. In other words, any CKY $p$-form on a metric
almost abelian Lie algebra is parallel for $2\leq p\leq n-2$. Moreover, we
characterize almost abelian Lie algebras admitting non-parallel CKY $p$-forms,
and we classify all Lie algebras with this property up to dimension $5$,
distinguishing also those cases where the associated simply connected Lie group
admits lattices.
