  Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory
are the Kapustin-Witten equations, a family of gauge-theoretic equations on a
four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to
a linear combination of two super charges in the twist. When $t=0$ and the
four-manifold is a compact K\"ahler surface, the equations become the Simpson
equations, which was originally studied by Hitchin on a compact Riemann
surface, as demonstrated independently in works of Nakajima and the third-named
author. At the same time, there is a notion of $\lambda$-connection in the
nonabelian Hodge theory of Donaldson-Corlette-Hitchin-Simpson in which
$\lambda$ is also valued in $\mathbb{P}^1$. Varying $\lambda$ interpolates
between the moduli space of semistable Higgs sheaves with vanishing Chern
classes on a smooth projective variety (at $\lambda=0$) and the moduli space of
semisimple local systems on the same variety (at $\lambda=1$) in the twistor
space. In this article, we utilise the correspondence furnished by nonabelian
Hodge theory to describe a relation between the moduli spaces of solutions to
the equations by Kapustin and Witten at $t=0$ and $t \in \mathbb{R} \setminus
\{ 0 \}$ on a smooth, compact K\"ahler surface. We then provide supporting
evidence for a more general form of this relation on a smooth, closed
four-manifold by computing its expected dimension of the moduli space for each
of $t=0$ and $t \in \mathbb{R} \setminus \{ 0 \}$.
