In the hat season, while complimenting someone about their hats, do we say
The hat looks good on your profile picture
Or
The hat looks good over your profile picture
Both sound similar to me. Please advice the correct way.
In your specific example
The hat looks good on your profile picture
would be the one to use since clearly the hat is placed within the boundaries of the profile picture
The hat looks good over your head on your profile picture
The hat looks good on your head in your profile picture
Would be longer ways of saying that the hat sitting on top of the user's head looks good
For the hat to be over the profile picture, it would need to be physically above the boundary of the picture even though technically the hat is placed above the picture in an image-layering sense
I'd say that on is a safe preposition to use. After all, in the same way we put paint on a canvas, we can superimpose an icon on a profile picture.
That said, over can be perfectly acceptable, too, particularly when the "hat" functions more like a mask than a hat.
Generally speaking, hats go on your head, and masks go over your face.
So, there's nothing wrong with saying either one, but which word works better might depend largely on how the hat looks, fits, and is designed to be worn. For example:
I might be inclined to say that the blue flight cap looks good on my profile picture (because of the way it sits atop the orange), but that the Darth Vader mask looks good over my profile picture (since it is worn over the front of the orange).
Even the preposition in might work, when the hat is used more like an adornment than as a hat or mask. For example, it wouldn't be incorrect to say that this Christmas tree works "in" my avatar picture quite well:
(Unless, of course, you think the tree looks ugly. In that case, you'd be lying – but your preposition choice would still be acceptable.)
over suggest movement (like we flew over Santiago) or when you cover a big surface. In this case it's just a punctual thing so I think on fits best here because it's just a part of a surface.