There were a few cartoons in which Bugs appeared without Elmer. They were "less successful" than the Bugs and Elmer ones, and without a "substitute Elmer" (i.e. Daffy Duck, etc.) they mostly were total flops.

There is something "missing" in the Bugs persona without a victim (i.e. a straight man) so the co-dependency is there, even if it's too subtle for some to see at first look.

John