This is just notes and stuff, sorry!
Harry now has a basilisk familiar, though no one knows it is a basilisk. As a familiar, their gaze and venom is only deadly to animals, unless their chosen human wills it otherwise.
Hagrid, not being a huge fan of snakes, allows Harry to keep it. Harry is granted special permission to do so after it is revealed he is a parselmouth and the two have already bonded.
Once born, the snake is only about a foot long, and it's favorite place to hide is in Harry's hair, seeming to shrink itself when wound through his locks.
He names her Selene due to the color of her eyes when she was born, and a pure white spot on the crown of her head shaped like a crescent moon.
She calls him Hairy Nest Mother.
No idea what Quirrelmort's reaction would be, but I imagine the man is confused cause after Harry learns no one can understand what he is saying when speaking in parseltongue, he does not hold back on the snark, and this Harry may mostly be a pure cinnamon roll, but he can also be a catty bitch, especially around Malfoy's and stupid people.
Later years, Selene playing matchmaker, trying to find her mother a nice mate, so she can have some brothers and sisters.