How is invisibility treated underwater? Does the water move around them? Can they be located by displacement? Is there anything that modifies concealment? Does how thick or murky have any effect? What if there is stuff in the water, like debris or seaweed?
An invisible underwater creature is treated as having concealment.
From the Aquatic Adventures book, p46, discussing how various magic effects behave differently in an underwater environment:
Invisibility: Underwater, an invisible creature still displaces water, making it possible to spot and target it as if it had only concealment, as per the description of invisibility on page 563 of the Core Rulebook. The invisibility bubble spells on page 60 hide the displaced water and thus allow effective invisibility underwater.
So a creature that uses invisibility underwater will be visible, but still benefits from concealment.
Or, the creature can use invisibility bubble instead and will not be visible when underwater.
As for whether the water is murky, that won't affect an invisible creature, although it will impose penalties on Perception checks to notice them:
Perception: Even in clear water, the way light travels underwater makes it harder to see underwater than on land, and water sometimes contains particulates that further obscure vision. Perception checks to see something are always made at a minimum –2 penalty underwater, and often a greater penalty when the water is murky (flowing water, such as in a current or river, is always at least somewhat murky).