He talks about a square matrix H(z) depending on a complex variable. He says that the eigenvalues of H(z) form a multi-sheeted Riemann surface (the multiset of E with det(E−H(z))=0 for some z), and explained that the spectrum has discrete (quantized) level for fixed z although the surface smoothly connects all eigenvalues.
This is just a fact of linear algebra over the complex numbers. The only connection to quantum mechanics is that for real z (if the coefficients are real analytic, as in his 2-dimensional examples) the matrix can be viewed as a Hamiltonian acting on a finite-dimensional Hilbert space.
Calling this quantization is just an illustration, not a hard fact; it cannot be used to quantize anything.