The point in a conversation where it all gets rather child-ish from both sides: A: That ball is pink. B: No, it isn't. Are you blind? It's obviously purple! A: No, it isn't! B: Yes, it is! A: No, it isn't, you blind c***. B: I am not a c***! A: Yes, you are, obviously, a c***! B: Am not! A: Are too! ...(etc.)
The conversation becomes a back-and-forth, a childish argument in this case. Merriam Webster lists synonyms that I feel are appropriate for the situation. Quarrel, bicker, spat, tiff, and squabble could also be words to describe this, but I'm more partial to back-and-forth because of the way the conversation goes. Both parties just shoot off the same tired argument and neither gives an inch.
I'd call it a "pre- ad hominem argument," or "the precursor to an ad hominem tirade," or something along those lines.
When interlocutors have abandoned rational categories, virtually any fallacy will do. The ol' ad hominem is a "logical" choice, I suppose. Here's a scenario which has its basis in a Saturday Night Live sketch ("Point/Counterpoint"?).
Dan Ackroyd: The president was clearly off base in his comments to the CEO.
Jane Curtin: He was clearly not off base. In fact, I suggest his comments were on target and quite apropos, given the circumstances.
Dan Ackroyd: I couldn't disagree with you any more strongly than I do.
Jane Curtin: Well, I couldn't disagree with you any more strongly than I do.
Dan Ackroyd: My level of disagreement with you is off the charts.
Jane Curtin: My level of disagreement with you is infinite.
Dan Ackroyd: Jane, you ignorant slut!