Weird movie I can’t remember

Hi all, my friends and I were talking about the weirdest movies we’ve ever seen and this one came to mind. From what I remember it was about a teenage boy who murdered a girl ( I think it was a classmate) and then sleeps with her dead body. This is seen by another girl who becomes enamoured with him. They somehow end up on the run or some kind of road trip together and she admits that she saw his crime and thought it was beautiful? (The nursery rhyme “do your ears hang low” is somehow relevant, she sings it when she catches him with the body maybe?) and I’m pretty sure the movie ends with her drowning.
Suffice to say it’s pretty weird and my mates were looking at me like I was crazy when I was describing it. Does anyone know what it’s called? Thanks