What is indexed monad?

As ever, the terminology people use is not entirely consistent. There's a variety of inspired-by-monads-but-strictly-speaking-isn't-quite notions. The term "indexed monad" is one of a number (including "monadish" and "parameterised monad" (Atkey's name for them)) of terms used to characterize one such notion. (Another such notion, if you're interested, is Katsumata's "parametric effect monad", ind