Unhinged OED or by approximate association Unglued
ADJECTIVE
Mentally unbalanced; deranged:
Etymology
Unhinged: "thrown into confusion," 1719, past participle adjective from unhinge.
Unhinge: recorded earlier in the mental sense of "to disorder" the mind, etc. (1612) than in the literal one of "to take (a door, etc.)
off its hinges" (1616); from un- (2) "opposite of" + hinge (n.).
Hinge as a verb meaning "to attach by a hinge" is recorded only from 1758.
Literally:
My husband was so furious the door came unhinged when he slammed it
Metaphorically:
My husband is unhinged!
Synonyms of confuse(d).
Applying it to the OP:
The two people arguing began connected to each other at some level. A healthy argument pursues a deeper level of intellectual or emotional unity on the dividing issue, but the ad hominem attack tears the door from its hinges, or unglues the coupled pair. From that point forward, their angry comments aggravate disunity rather than promoting unity. Furthermore, that breech in the friendship tears each individual apart inside. As they unhinge internally, their argument unhinges externally.
Somebody needs to stop the argument and put the doors back on their hinges!