An eye or eyes

For Japanese people, I think plural is one of the most difficulties regarding English. In Japanese language, we rarely use plural. Even if you ate two apples, you should say "I ate apple." in Japanese. Even if you see many flowers in a garden, you should say "What beautiful flower."
(By the way, an article doesn't exist in Japanese.)

I felt strange when I saw some English idiom using "eye."
For example;

"Her beautiful hair caught my eye."
 Catching only one eye, not both eyes?

"Keep your eyes peeled for new information."
"Keep an eye out for new information."

Both are almost the same meaning, but the first sentence uses "one's eye" and the second one uses "an eye."

I think that it's much better to use both eyes when you try to pay attention something. haha.