The harmonic oscillator with non-positive coupling constant, i.e., $H_0 f = -f''(x) + cx^2f(x)$, with $c \in \mathbb{C} \setminus [0,+\infty)$, is non-self-adjoint (see the book in preparation by Sjöstrand (http://sjostrand.perso.math.cnrs.fr/161014bok.pdf)). So in your case take $c = -1$, then the spectrum is purely imaginary.

The harmonic oscillator with non-positive coupling constant, i.e., $H_0 f = -f''(x) + cx^2f(x)$, with $c \in \mathbb{C} \setminus \mathbb{R}$, is non-self-adjoint (see the book in preparation by Sjöstrand (http://sjostrand.perso.math.cnrs.fr/161014bok.pdf)). This does not answer the question since $c = -1$ is not allowed (and gives a symmetric operator).