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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 Eigenvalue statistics of the real Ginibre ensemble. 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 $\beta=1,2,4$.