In lecture 3 (the most relevant explanations are between 46:00 and 56:30) of his Mathematical Physics course, Carl Bender mentions that the evaluation of complex functions on different Riemann sheets can describe the quantization of physics observed in nature.

How does this "quantization method" work exactly, or is it just a different (more mathematical) point of view to look at things?

How is this way of thinking about quantization linked to the usual methods of quantization in physics, such as for example covariant quantization, light-cone quantization, or BRST quantization?