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Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting I recommend the books of Bolibrukh.

Applications of compact Riemann surfaces to solitons ("Explicit solutions" of the Koreweg-de-Fries equation etc. In a comprehensive course of algebraic curves and Riemann surfaces taught by Drinfeld, that I took in early 1980-s this was included as an example of application.

Belyi theorem was used in this course as a HW exercise, but since then much interesting stuff was added to this.

Myself, I use holomorphic dynamics to "spice" my Riemann surface courses, also Kleinian groups.. Especially Sullivan's proofs of the finiteness and non-wandering theorems.