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I sit at a screen 8-16 hours a day. I get strong headaches every single day, for which I keep a supply of ibuprofen at work and home. 400mg a day is generally plenty.
I do not wear my glasses while at the screen, as it's close enough that I don't have any issues.
I suspect my headaches are neck muscle related, not eyesight, but I haven't investigated further.
Maybe try e-ink? There are e-ink monitors on the market if that works, if the problem's the light. You can also try software like Redshift[1] for regular monitors.
If it's your neck, at least put some books under your monitor stand, if you're not able to get better monitor mounts. Or lower your chair. You should be looking straight ahead for your monitor, so your neck muscles shouldn't have to do much work.
Definitely not a good plan, especially Ibuprofen. My mother took Ibuprofen for years to manage her arthritis and that absolutely wrecked her kidneys. Not to mention Ibuprofen can also cause internal bleeding. It's a quick short term fix but not a good long term solution.