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.