Complex complex landscapes

We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the B\'ezout bound $\log\overline{\mathcal N}~N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.