I'll keep my eye out for you - to keep looking for someone like they're going to give you t

to keep looking for someone like they're going to give you trouble

I'll keep my eye out for you, or I'll beat you in the final exam.

I'll keep my eye out for you, or I'll show up at your house at 10 p. m.

I'll keep my eye out for you, or I'll make your life a nightmare.