  We provide new information concerning the pseudospectra of the complex
harmonic oscillator. Our analysis illustrates two different techniques for
getting resolvent norm estimates. The first uses the JWKB method and extends
for this particular potential some results obtained recently by E.B. Davies.
The second relies on the fact that the bounded holomorphic semigroup generated
by the complex harmonic oscillator is of Hilbert-Schmidt type in a maximal
angular region. In order to show this last property, we deduce a
non-self-adjoint version of the classical Mehler's formula.
