The rapid twisting and running of a hunted hare is here used as a metaphor for the pursuit of a topic in an animated conversation, especially one in which the participants hold strong views.
RELATED IDIOMS :
mad as a March hare
completely crazy – informal
In this expression, a hatter refers to Lewis Carroll's character, the Mad Hatter, in Alice Adventures in Wonderland (1865). It is thought that hatters suffered from the effects of mercury poisoning because of the fumes arising from the use of mercurous nitrate in the manufacture of felt hats. The March hare version refers to the way hares leap about during the breeding season.
run with the hare and hunt with the hounds
try to remain on good terms with both sides in a conflict or dispute – British
This expression has been in use since the mid 15th century.