  We explore the possibility that physics at the TeV scale possesses
approximate $N = 2$ supersymmetry, which is reduced to the $N=1$ minimal
supersymmetric extension of the Standard Model (MSSM) at the electroweak scale.
This doubling of supersymmetry modifies the Higgs sector of the theory, with
consequences for the masses, mixings and couplings of the MSSM Higgs bosons,
whose phenomenological consequences we explore in this paper. The mass of the
lightest neutral Higgs boson $h$ is independent of $\tan \beta$ at the tree
level, and the decoupling limit is realized whatever the values of the heavy
Higgs boson masses. Radiative corrections to the top quark and stop squarks
dominate over those due to particles in $N=2$ gauge multiplets. We assume that
these radiative corrections fix $m_h \simeq 125$ GeV, whatever the masses of
the other neutral Higgs bosons $H, A$, a scenario that we term the $h$2MSSM.
Since the $H, A$ bosons decouple from the $W$ and $Z$ bosons in the $h$2MSSM at
tree level, only the LHC constraints on $H, A$ and $H^\pm$ couplings to
fermions are applicable. These and the indirect constraints from LHC
measurements of $h$ couplings are consistent with $m_A \gtrsim 200$ GeV for
$\tan \beta \in (2, 8)$ in the $h$2MSSM.
