  Donaldon constructed a hyperk\"ahler moduli space $\mathcal{M}$ associated to
a closed oriented surface $\Sigma$ with $\textrm{genus}(\Sigma) \geq 2$. This
embeds naturally into the cotangent bundle $T^*\mathcal{T}(\Sigma)$ of
Teichm\"uller space or can be identified with the almost-Fuchsian moduli space
associated to $\Sigma$. The later is the moduli space of quasi-Fuchsian
threefolds which contain a unique incompressible minimal surface with principal
curvatures in $(-1,1)$.
  Donaldson outlined various remarkable properties of this moduli space for
which we provide complete proofs in this paper: On the cotangent-bundle of
Teichm\"uller space, the hyperk\"ahler structure on $\mathcal{M}$ can be viewed
as the Feix--Kaledin hyperk\"ahler extension of the Weil--Petersson metric. The
almost-Fuchsian moduli space embeds into the
$\textrm{SL}(2,\mathbb{C})$-representation variety of $\Sigma$ and the
hyperk\"ahler structure on $\mathcal{M}$ extends the Goldman holomorphic
symplectic structure. Here the natural complex structure corresponds to the
second complex structure in the first picture. Moreover, the area of the
minimal surface in an almost-Fuchsian manifold provides a K\"ahler potential
for the hyperk\"ahler metric.
  The various identifications are obtained using the work of Uhlenbeck on germs
of hyperbolic $3$-manifolds, an explicit map from $\mathcal{M}$ to
$\mathcal{T}(\Sigma)\times \bar{\mathcal{T}(\Sigma)}$ found by Hodge, the
simultaneous uniformization theorem of Bers, and the theory of Higgs bundles
introduced by Hitchin.
