Who says half of his eyes were asleep and half awake?
While a number of sources refer to his eyes taking sleep in turns, the only source I can find that specifies how many slept at once is Ovid:
Argus, Aristorides, whose head was circled with a hundred glowing eyes; of which but two did slumber in their turn whilst all the others kept on watch and guard.
Ovid's Metamorphoses I, 622-641
Also, when he is being lulled to sleep by Hermes, more of his eyes close gradually, rather than as a unit:
but Argus strove his languor to subdue, and though some drowsy eyes might slumber, still were some that vigil kept.
Ovid's Metamorphoses I, 682-688