OK, so this is an interesting question. To start, I’m a circadian researcher, specifically focusing on how bacteria can influence our circadian clocks.

It is indeed correct that most animals utilize the sky (specifically, the ambient brightness) to determine the time. But the circadian clock is incredibly entrenched. It evolved ages ago, and so by extension, virtually every single animal inherited the same circadian clock (with some modifications). Animals as distinct as fruit flies, fish, and humans have similar circadian clocks. And the circadian clock is unbelievably important, more than people give it credit for. Night and day are incredibly different environments, and every single animal needs to be able to predict and accommodate for the cold that comes at night and the UV radiation that comes during the day. And there’s a plethora of other, subtle changes to the environment that we don’t fully understand yet. For instance (and probably a bit unsurprising in hindsight), the population of bacteria in the air changes at day versus at night. Soil bacteria, for some reason, act differently at day versus at night. Presumably plants (which are definitely circadian) are influencing the soil bacteria in some way through their roots, but it’s not entirely clear.

An interesting consequence of the importance of the circadian clock is that animals have evolved multiple, redundant ways of telling time. If you lock a person in a dark box for weeks to months (scientists have performed this experiment in the past), the circadian clock running within the person is still able to somewhat-accurately tell the time, and we can use experiments like this to tease apart how the circadian clock utilize different cues to figure out the right time. One of the more interesting cues (and the one that I research) is how our bodies use surrounding bacteria to tell the time. And it’s known that eating food affects the circadian clock. Food availability, of course, is pretty circadian, especially if you eat food that is circadian. If a mouse comes out at night, then there’s no point hunting for mice in the day. I’m not too knowledgeable about deep sea animals (I’m really more focused on mammals), but a quick literature search suggests that deep sea animals do have circadian clocks, inherited from the same ancient ancestors that we got ours from. The conclusions appear to be similar to what I’ve said above - namely, that even if there’s no sky, having a functional circadian clock is necessary simply because other things, such as food, are themselves circadian.

So my answer is, presumably, that deep sea animals can already accurately tell the time, presumably by keeping track of when they last ate

I’m guessing a heavy pendulum still works under water. Heavy enough that the effects of drag get negligiable. As do springs.

Although fluid dynamics often surprise me.

One can make a clock out of either of those.

A few uneducated thoughts:

I feel like there should be a way to use tides to track the passage of time. Maybe by monitoring the shifting water at fixed points?

While a clepsydra may not work underwater, a inverse one that slowly releases air instead of water might. Similarly, any container that slowly releases sand, or other such substance, at a constant rate could be used. Either by adjusting a scale, releasing counterweights, or even tracking how much has been released into another container (like an hourglass).

For measuring brief periods of time, it could be how long it takes bubbles to reach a particular height from the sources. I’m sure it would be a bit different at each depth/pressure, so you’d need to take that into account.

I don’t think I agree with the premise. Either time changes something and that makes tracking time important to me as a sea creature… And I’d use that as my reference. Or time is a meaningless concept. And I wouldn’t bother. I mean humans in downtown city also don’t keep track of the tides in some arbitrary place. It’s superfluous.

I would use the slow movement of sea snails. I’d have an enclosed area with a sea snail on one end and food on the other end to lure it, and would treat every few inches as an hour (because that’s what it would be).

(The following content is fiction)

Train my brain to interpret the EMI of (older) submarine communications cables by using my lateral line organ.

After that step is completed it would be trivial to implement basic protocols, such as NTP or ICMP.

Temperature of the water? Very inconsistent and tough to manage I’d guess

Or smth to do with tides and currents. (I have no idea how any of that works 😭)

Using the erosion of some rock that is consistent to mark time? That would be useful for months or years but probably too inaccurate to calculate seconds.

Rely on your good old Biological clock