Wow this potentially has a long answer! The short answer is that it's a reference to the book "Alice's Adventures in Wonderland", which begins with Alice falling down a rabbit-hole that turns out to be extremely deep and full of magical, nonsensical stuff. You can probably work out for yourself how that relates to taking pills! I wouldn't say this is a common phrase in English. It is just an allusion to a book.

Good expression! It is an Alice in Wonderland reference. The rabbit hole refers to some kind of adventure or ordeal that is lengthy and has many twists and turns, and is potentially difficult to get out of. It is usually used in a negative sense, e.g "You talked to him about his conspiracy theory? I was afraid to go down that rabbit hole?" Meaning the speaker thinks conversation was probably going to be long and uncomfortable and difficult to get out of once started.