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

2. 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. 

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

4. 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.