Answer
Arguments can be mean and hurtful.
If it wasn't for the fact that both of the people were saying mean things, it could count as bullying.
The phrase "Anger is just a way of expressing fear" is true.
That person could be hurt at your words, as much as you are hurt by his/her words.
The best option is to stop talking and run away.
If you're talking about a debate argument, then it is important to be respectful.
It could provide a big stain on your reputation.