First of all, it is sufficient to consider only one parameter: making a change of the independent variable $z\mapsto kz$, with appropriate $k$ one can eliminate either $a$ or $b$. Let us eliminate $b$ and consider $$-y''+(x^4+ax^2)y=\lambda y.$$ Then eigenvalues (in $L^2(R)$) become functions of $a$, and the asymptotics is $$\lambda_n\sim cn^{4/3},$$ where $c$ is an absolute constant (it does not depend on $a$). One can write several terms of asymptotic expansion in decreasing powers of $n$, coefficients of these further terms will depend on $a$. In fact one can write and prove an infinite asymptotic expansion, but the resulting series is divergent. These functions $\lambda_n(a)$ were very much studied, I mention first of all the paper:
C. Bender and T. Wu, Anharmonic oscillator. Phys. Rev. (2) 184 1969 1231–1260.
This paper has more than 1000 references on Google scholar. By looking in these references you can obtain a more or less complete picture of what is known. (The situation is roughly like this: all statements in this paper are correct, and it contains a very complete discussion of these eigenvalues. However most of the things are proved on the "physical level of rigor", or just illustrated by computation. In the subsequent papers most of these statements were rigorously justified.) If you are only asking about asymptotic expansion for real $a$, it is written explicitly in Bender-Wu, and it is easy to justify. The difficult thing is proving the global analytic properties of $\lambda_n(a)$, for complex $a$. These functions are very complicated, indeed. Some principal mathematical papers which discuss them are:
B. Simon, Coupling constant analyticity for the anharmonic oscillator, Ann. Phys., 58 (1970) 76–136.
A. Voros, The return of the quartic oscillator, Annales de l'Institut Henri Poincare. Section A, Physique Theorique; v. 39(3); 1983, p. 211-338.
B. Simon, Large order and summability of eigenvalue perturbation theory: a mathematical overview, Intl. J. Quantum Chemistry, 21 (1982) 3–25.
E. Delabaere, F. Pham, Unfolding the quartic oscillator, Ann. Physics 261 (1997), no. 2, 180–218.
J. Loeffel and A. Martin, Proprietes analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade, dans le livre: Cargese Lectures in Physics, vol. 5, Gordon and Breach NY, 1972, 415–429.
Kwang C. Shin, Schrodinger type eigenvalue problems with polynomial potentials: asymptotics of eigenvalues, arXiv:math.SP/0411143v1.
A. Eremenko and A. Gabrielov, Analytic continuation of eigenvalues of a quartic oscillator, Comm. Math. Phys., v. 287, No. 2 (2009) 431-457.