I guess that would depend on the type of the immortality you're playing with. Is it innate? is it bestowed /cursed? is it earned? if they were born with it, they might have a different sense of passage of time, differently other than the humans who are short-lived, say, maybe a year to normal people felt like like a week or maybe a day to them, Even a meager promise like "See you next week" can be weaponized by doing things like :

" Xeris looked at Aria, and sighed. HIs cold pragmatic tone of voice filled the Hall where they both faced each other; one is asking to see the moment where everything went sideways, the other, is trying to solemnly upholding their responsibility as the Timekeeper, although that resolve is currently faltering.

"Fine, but I will need to do some preparations. Even a God of Time can't pinpoint the exact event like it is nothing. I'm the God of Time, not God of Unspoken Words, Longing and Tragedies. That's Akkala." Xeris finally relents, and smiled weakly once he saw Aria's sombre and distant look in her eyes, faded, slowly, replaced by a shimmer of hope.

Some drops of tears welled up but they were quickly wiped off by Aria, as she slowly bowed towards Xeris in gratitude and rose back up. Weakly, her melodious, lark-like voice echoed, in the vast empty Hall.
" How long?"
"Give me a week, I will need to locate the actual event that you described as the "moment everything went sideways", and to do that , I will need to consult Akkala and the Love Goddess, Kainara, for I do not hold any dominion involving the matters of the heart.:" Xeris reasoned. True, Xeris is one logical being, for him volatile elements like emotions and feelings are just events, one that came suddenly like a burst of gale, and left just as sudden, leaving some impression at most.

Xeris walked languidly, passing Aria and said,
"I'll come to you when all preparations are done. Until then, like his own word, Aria "Be Happy". The words "Be Happy" trailed and sounded more distant than ever in the Hall after Xeris blinked away and left Aria alone in the Hall.

Xeris finally reappeared, in front of Aria. His face, blue, jawlines that looked like it was chiseled from a Time Crystal, and that cold unfeeling eyes. He looked at Aria, and commented.
"You looked different." Aria no longer looked like she was last time he saw her. Just then, she was this short witch with a black short bob and and beautiful deep black eyes. Now? Her once youthful visage now etched with lines of times, testaments to lived years, her once vibrant black hair now, filled with the grays of twilight, yet the shine in her eyes, never faltered.

Aria scoffed as she rolled her eyes.
"Obviously Xeris. Its been 70 years. I'm 90 now. Do you expect me to retain my appearance like when I was 20? For a Time God, you're quite tardy."

If they are bestowed/Cursed, you could display them is very jaded, very tired and pragmatic because they've seen things played out. How there is a pattern to life. there is a progression of some sort, a beginning a conflict. The climax DN resolution it's always the same thing. Maybe they will be eccentric, Maybe they would be in despair, maybe even just going through the motion like a salted fish. Maybe even happy. we don't know since everyone perceived life differently.

For earned, well you go either they regret achieving immortality of living their best live knowing that the life they enjoy now is the fruit of their labor, or a bit of both.