  We study the eigenvalue problem $-u"+V(z)u=\lambda u$ in the complex plane
with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity
along the two rays $\arg z=-\frac{\pi}{2} \pm \frac{2\pi}{m+2}$, where
$V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most
$m-1\geq 2$. We provide an asymptotic formula for eigenvalues and a necessary
and sufficient condition for the anharmonic oscillator to have infinitely many
real eigenvalues.
