have/keep half an eye on

:  to retain in one's possession or power

:  to refrain from granting, giving, or allowing

:  to have in control

:  the means or provisions by which one is kept

:  one that keeps or protects: such as

:  fortress, castle

:  either of two equal parts that compose something

:  a part approximately equal to one of these

:  one of a pair: such as

:  being one of two equal parts

:  that is approximately equal to either of two equal parts that compose something :  amounting to approximately half

:  falling short of the full or complete thing :  partial

:  in an equal part or degree

:  not completely :  partially

:  by any means :  at all

:  a specialized light-sensitive sensory structure of animals that in nearly all vertebrates, most arthropods, and some mollusks is the image-forming organ of sight

:  the nearly spherical usually paired hollow organ of sight in vertebrates that is filled with a jellylike material, is lined with a photosensitive retina, and is lodged in a bony orbit in the skull

:  all the visible structures within and surrounding the orbit and including eyelids, eyelashes, and eyebrows

:  to fix the eyes on :  look at

:  contemplate, consider

:  to watch or study closely