A very well known result in random matrix theory is that there exists "nice" (i.e., with independent entries) tridiagonal matrix for the $\beta$-ensembles of random matrix theory $$\propto\prod_{i<j}|\lambda_j-\lambda_i|^\beta\exp\left(-\frac{\beta}{4}\sum_{i}\lambda_i^2\right),\qquad\lambda_i\in\mathbb R\tag{1}$$ extrapolating between the classical values of $\beta=1,2,4$ given by the GO/U/SE (see this paper by Edelman and Dumitriu). This discovery subsequently paved the way for many interesting results regarding the behaviour of particle systems described by $(1)$, such as those related to the stochastic Airy operator.

It is often said that, from the point of view of statistical physics, the interaction term $|\lambda_i-\lambda_j|$ in the joint density $(1)$ is not natural in a one-dimensional setting, and that a more "appropriate" model for this type of repulsion would be in two dimensions: the "$\beta$-Ginibre" ensemble, given by $$\propto\prod_{i<j}|\lambda_j-\lambda_i|^\beta\exp\left(-c(\beta)\sum_{i}|\lambda_i|^2\right),\qquad\lambda_i\in\mathbb C.\tag{2}$$

After searching in the literature, I cannot find "nice" matrix models for $(2)$ that would extend the case $\beta=2$ for i.i.d. complex Gaussian models (though, as noted by Carlo Beenakker, the density $(2)$ for $\beta=1,4$ is not given by the real or quaternion i.i.d. Gaussian model).

Is it known that such models exist, but are somehow not very interesting in terms of applications, or are there good reasons to believe that such a model does not exist? Of course we cannot expect a self-adjoint tridiagonal model, but it seems to me a natural question wether or not this can be done and if there might be interesting applications.