Summary: We study asymptotics of orthogonal polynomial measures of the form \(|\mathcal H_N|^2d\gamma\) where \(\mathcal H_N\) are real or complex Hermite polynomials with respect to the Gaussian measure \(\gamma\). By means of differential equations on Laplace transforms, interpolation between the (real) arcsine law and the (complex) uniform distribution on the circle is emphasized. Suitable averages by an independent uniform law give rise to the limiting semi-circular and circular laws of Hermitian and non-Hermitian Gaussian random matrix models. The intermediate regime between strong and weak non-Hermiticity is clearly identified on the limiting differential equation by means of an additional normal variable in the vertical direction.