Hermitian operator on a nonorthogonal two-state basis

For normalized independent states with $H|\psi ⟩=g|\varphi ⟩$ and $H|\varphi ⟩=g^{\ast }|\psi ⟩$, Hermiticity is equivalent to $g⟨\psi |\varphi ⟩\in \mathbb{R}$. Writing $r=g⟨\psi |\varphi ⟩/|g|$, the orthonormal eigenstates are proportional towith eigenvalues $\pm |g|$ and squared norms $2(1\pm r)$.