I had this problem at my last job -- banks of lights meant that you couldn't just turn off your overhead light (and we weren't allowed to remove individual tubes). Your decision affects other people, so you have to work with them to find a solution that makes everybody happy.
If your coworker's problem is only that she would be left in the dark, then get her a lamp! Or get a lamp that provides light for both of you, like a pole lamp (what we did in my case). Note that your employer might not be willing to pay for this; from their perspective they gave you light and you rejected it. But ask; there might be others with this issue and they may see getting a few lamps as a small cost to keep people productive. If not, buying a lamp or two for you and your coworker should be a small cost to you, given what's at stake.
If she actively likes the fluorescent light then you have a bigger problem; you are at odds with respect to your shared environment, whether it's light, the window blinds, heating, or something else. If you can't compromise then you should ask if you can move to another desk. Make it easy for your manager (or office manager, or whomever decides these things at your workplace): scout out a location that would work better for you, e.g. with somebody who also doesn't like the fluorescent lights or doesn't mind replacing them with a lamp. It's best if your proposed location is not seen as "better" than your current one; you're not trying to move up, just out.
Finally, if there is a documentable health issue like severe headaches, this can give you more leverage with your employer in some countries. But don't use that unless you have to. Start by assuming that everybody involved is a reasonable person; naturally they don't want you to get headaches at work, and naturally you don't want to inconvenience your coworkers or gain special privileges like that desirable window seat. Look for the smallest change that fixes the problem and you'll be seen as a team player.