Even for $\beta=1$, so for a real matrix, the eigenvalue distribution in the Ginibre ensemble does not have the form (2), so this "interpolating distribution" is not a natural object for complex eigenvalues. For the Ginibre eigenvalue distribution at $\beta=1,2,4$ see equations 1,2,3 of <A HREF="http://arxiv.org/abs/0706.2020">Eigenvalue statistics of the real Ginibre ensemble</A>. As you can see, it has a completely different form for these three values of $\beta$, so there is no natural notion of an "interpolation" --- unlike in the case of real eigenvalues, where it has the same form (1) for $\beta=1,2,4$.