I think Dr. Bishop suggested that Peter isn't his biologocal son-he is some kind of experiment like that man from second episode, but Dr. is a little crazy so he could just imagine something strange about his son.

Must be an illness or similar that Walter managed to get rid of with his research and tests on Peter, but perhaps it didn't go to plan...?

I have a feeling that Walter experimented on Peter.

I think he is a clone like that other guy and he just doesnt need anything

I think Peter's a clone

Do to Walters apparent instability plus a conversation I seem to remember where Peter came right out and said something about being the subject of an experiment or something like that it's where I'd place my bet.
As for checking his eyes there are a number of things he could be checking for such as blood problems, chemical imbalances do to organ malfunction, or brain abnormalities and more.
I think it was placed in our minds so they can bring it up at a later time.