  We study moduli spaces of solutions to the extended Bogomolny equations on
$\Sigma \times \mathbb{R_{+,y}}$ with gauge group
$\operatorname{SL}(2,\mathbb{C})$ satisfying the generalized Nahm pole boundary
condition as $y\to 0$ and limiting to complex flat connections as $y\to
\infty$. Refining the Kobayashi-Hitchin correspondence of \cite{MH2}, we
identify these moduli spaces with certain holomorphic lagrangian sub-manifolds
inside the moduli space of Higgs bundles.
