I went to a restroom today and my eyes started burning and watering, and it was from fumes from some kind of cleaner, bleach or ammonia, but I think it was bleach. Is there any reason to be alarmed?

No, you should be fine. I get the same thing at work every morning. The person who cleans our office is leaving just as I get in early, and the kitchen & bathrooms always are a little strong of bleach smell.
Just don't linger in there longer than you have to. Or next time, you can do like I do and cover my mouth & nose with my shirt till I get out. And then get some fresh air if you can.

Okay, thanks. I was just wondering because it was very strong and I was concerned if it hurt my eyes. I must have been in the room for no longer than a few minutes.