Salam Asif tells me of one such moment has been immortalised in a verse between the two (recorded by Nadvi).
While the couple sat silently by the river bank, basking in the serenity of Yamuna; Shah jehan observed the raging waves lashing upon the river-bank rocks; and then he looked at his beloved wife’s beautiful face – and slowly uttered an impromptu poetic line in homage to her beauty:
آب از برائے دیدنت می آید از فرسنگھا
Aab -az-braai deednat me aayed az farsang ha
(trans: See, how these waves travel to this point from miles afar; to have a glimpse of your majestic face)
Mumtaz Mahal responded instantly composing the next poetic line in the same metre to complete the couplet:
از ہیبتِ شاہِ جہاں سر می زند بر سنگ ھا
Az haybat-e-Shah-e-jahaan sar mi zanad br sang ha
(tran: But look how these waves crush their mighty crests (heads) against the rocks, at being overwhelmed by the majesty of my emperor Shah-e-Jahan’s presence here).
Note: this photo is from the side of the Taj Mahal of course they would have sat in the Agra Fort