Diagnosis
No, do not use Windex, glass cleaner, or any cleaner with ammonia, alcohol, or abrasive chemicals. The screen has a delicate anti-glare coating that those cleaners will strip right off, leaving permanent cloudy or streaky damage.
How to Fix It
The right way is simple and gentle. 1) First, power the monitor off and let it cool. 2) Use a dry, soft microfiber cloth (like the kind for cleaning glasses or camera lenses) to gently wipe away dust. 3) For fingerprints or smudges, lightly dampen a corner of the microfiber cloth with distilled water only. Tap water can leave mineral deposits. 4) Wipe the screen gently in a circular motion, then immediately dry/buff it with the dry part of the cloth. 5) Never spray liquid directly onto the screen—always onto the cloth.
If That Didn't Work
If you have a stubborn spot, you can use a 50/50 mix of distilled white vinegar and distilled water as a slightly stronger cleaning solution. Again, apply a tiny amount to the cloth, not the screen. Test it on a small corner first. Commercial screen cleaners labeled as safe for monitors with anti-glare coatings are also okay, but water is usually sufficient.
Heads Up
The anti-glare coating gives the screen its matte, non-shiny finish. If you damage it with harsh chemicals, the damaged spots will look shinier and more reflective than the rest of the screen, and there's no way to fix it. When in doubt, gentler is always better.