The phrase "bat the eye" is a commonly used idiom that means to quickly close and open one's eyelids, often in surprise or confusion. However, there are several synonyms that can be used to convey a similar meaning or emotion. "Flutter the eyelashes" refers to the rapid movement of the eyelashes instead of the eyelids. "Blink" is a more general term that can refer to any rapid closing and opening of the eyes. "Wink" specifically refers to closing one eye quickly, often as a gesture of secret communication or flirtation. "Nod" can also be used to indicate a quick and subtle gesture of agreement or acknowledgement, often with the head instead of the eyes.