  A new non-Hermitian E2-quasi-exactly solvable model is constructed containing
two previously known models of this type as limits in one of its three
parameters. We identify the optimal finite approximation to the double scaling
limit to the complex Mathieu Hamiltonian. A detailed analysis of the vicinity
of the exceptional points in the parameter space is provided by discussing the
branch cut structures responsible for the chirality when exceptional points are
surrounded and the structure of the corresponding energy eigenvalue loops
stretching over several Riemann sheets. We compute the Stieltjes measure and
momentum functionals for the coefficient functions that are univariate weakly
orthogonal polynomials in the energy obeying three-term recurrence relations.
