  We study resolutions of the rational map to the moduli space of stable curves
that associates with a point in the Hitchin base the spectral curve. In the
rank two case the answer is given in terms of the space of quadratic
multi-scale differentials introduced in [BCGGM3]. This space defines a
compactification (of the projectivization) of the regular locus of the
$\mathrm{GL}(2,\mathbb{C})$-Hitchin base and provides a compactification of the
Hitchin system by compactified Jacobians of pointed stable curves. We show how
the classical $\mathrm{GL}(2,\mathbb{C})$- and
$\mathrm{SL}(2,\mathbb{C})$-spectral correspondence extend to the compactified
Hitchin system by a correspondence along an admissible cover between
torsion-free rank $1$ sheaves and (multi-scale) Higgs pairs of rank $2$.
