The Higgs branch is a hyperkähler manifold as was shown by Luis Alvarez-Gaume and Daniel Freedman in their 1981 paper Geometrical Structure and Ultraviolet Finiteness in the Supersymmetric Sigma Model. Including gravity the supersymmetry becomes local. Then one needs to add the same Hodge condition to the special Kahler Coulomb branch as in the N=1 case. Jonathan Bagger and Edward Witten demonstrated in their 1982 paper Matter Couplings in N=2 Supergravity that in this case, the Higgs branch must be a quaternionic Kähler manifold.