For all its bugginess I really like the new Django Console. I noticed there are two menu buttons. The close window button and the stop button. The stop button is completely pointless b/c when you stop it there is nothing to do but close the window (you can't recover from it), and if you close the window it is stopped. So... Could you change the stop button to be simply a disconnect. Effectively doing this.
from django.db import connections, DEFAULT_DB_ALIAS
connection = connections[DEFAULT_DB_ALIAS]
This way you can keep the session going.