Are you going to use the window seat as a seat? If yes then do an inside mount roman shade. If no you can hang a curtain from the ceiling to the seat

Roman shades

I have curtains on my window seat -- it doesn't seem odd at all to me.

Do you have other windows in the room?

Plantation shutters are another option. Browse through images after googling “window seat”, and see which look you like the best. (Most have shades.)

We did plantation shutters over our window seat.

I hate plantation shutters a nice roman shade in a fabric that works with your bedding

Here's one with both shutters and curtains. However, my preference would be just shutters or a Roman shade.

Looks as though you may need some privacy at different times of the day? I would consider plantation shutters, a simple slant will give you privacy without changing overall look.

Thanks so much for the feedback! This is super helpful. Does anyone know if there's a place where you can buy Roman shades without going through a custom window covering company? Or do they need to be custom made because of window size?

What is the window size?