The phrase "in the bat of an eye" is an idiom used to describe something that happens very quickly or suddenly. There are many phrases that can be used as alternatives, including "in a snap," "in a flash," "in a jiffy," "in a twinkling," and "in no time at all." Another option is "in the blink of an eye," which is similar to the original phrase but uses a common physical action as a reference point. These synonyms can be used to add variety to writing or speech, or to convey a sense of urgency or surprise.