No, you're thinking of the caduceus, or kerykeion, which is in fact a symbol of Hermes. Like Hermes' famous sandals, the caduceus has wings on it. It also has two snakes rather than one. The snakes were originally shown as ribbons, which were a typical sign of a herald in the ancient world (Hermes is, after all, the messenger god). In the early medieval period this double-snaked staff was an alchemical symbol; eventually it became a symbol of medical science in the light of a medical practitioner's capacity to heal or harm, an indication that in hopeless cases he had the ability to kill a patient through euthanasia.