  Any quaternionic K\"ahler manifold $(\bar N,g_{\bar N},\mathcal Q)$ equipped
with a Killing vector field $X$ with nowhere vanishing quaternionic moment map
carries an integrable almost complex structure $J_1$ that is a section of the
quaternionic structure $\mathcal Q$. Using the HK/QK correspondence, we study
properties of the almost Hermitian structure $(g_{\bar N},\tilde J_1)$ obtained
by changing the sign of $J_1$ on the distribution spanned by $X$ and $J_1X$. In
particular, we derive necessary and sufficient conditions for its integrability
and for it being conformally K\"ahler. We show that for a large class of
quaternionic K\"ahler manifolds containing the one-loop deformed c-map spaces,
the structure $\tilde J_1$ is integrable. We do also show that the
integrability of $\tilde J_1$ implies that $(g_{\bar N},\tilde J_1)$ is
conformally K\"ahler in dimension four, but leads to an extremely stringent
constraint in higher dimensions that we show is never satisfied by a
quaternionic K\"ahler manifold in the image of the one-loop deformed
supergravity $c$-map. In the special case of the one-loop deformation of the
quaternionic K\"ahler symmetric spaces dual to the complex Grassmannians of
two-planes we construct a third canonical Hermitian structure $(g_{\bar N},\hat
J_1)$. Finally, we give a complete local classification of quaternionic
K\"ahler four-folds for which $\tilde J_1$ is integrable and show that these
are either locally symmetric or carry a cohomogeneity $1$ isometric action
generated by one of the Lie algebras
$\mathfrak{o}(2)\ltimes\mathfrak{heis}_3(\mathbb R)$, $\mathfrak{u}(2)$, or
$\mathfrak{u}(1,1)$.
