An excellent expression. To keep an eye, or to keep an eye on things means to maintain a watchful eye. You monitor what is happening with an appropriate zeal to the occasion. Another idiom would be, to keep an eye peeled. It means you are watching and protecting something, or someone.
keep one's eye on the prize an eye for an eye sleep with one eye open keep your eye on the ball give someone the evil eye hit the bull's eye have a bird's eye view get some shut eye (sleep) turn a blind eye to