Bsrgvty

I was thinking of an alternate history, in a part of a civilization, living along the Eastern Mediterranean, during the Bronze Age moves to the sea floor to found an underwater civilization. They build underwater homes on the sea floor, and use buckets to transport air from the surface of the Mediterranean to their underwater homes. They use sand to desalinate the water to drink. Over several thousand years this underwater civilization spreads throughout the Mediterranean and beyond.

Would this underwater civilization be possible during the Bronze Age? If so what would be the best way for the members of this civilization to bring air to breath when traveling underwater?

Would this underwater civilization be possible during the Bronze Age?

No.

First thing off:

They use sand to desalinate the water to drink.

Sand might be good at filtering out particles suspended in water. But salt is present in water as a solution, sand does nothing to it. Else every beach would be as good as a fresh water lake.

They build underwater homes on the sea floor

Also this doesn't stand: first, lack of materials and technologies to build an underwater dome, second and even more important, how do you keep a fire in an underwater environment with no air recirculation?

they use buckets to transport air from the surface of the Mediterranean to their underwater homes

To pull underwater a 10 liter bucket you need to pull 100 N, or 10 kg: without proper ballast and swimming equipment it's an hassle, believe me. And that would be the air you would use with just about 5 breaths. Lacking other method, this one is absolutely inefficient.

Wouldn't it make more sense if they just migrated to a tunnel network that was underground but had sea access. As in tunnels that terminate at caves and those caves have pools in them with flooded tunnels leading out to the sea.

Then perhaps your people could develop something like a breathing helmet made of leather sealed with pitch or resin or similar allowing them to move through the tunnels. They wont be able to see of course but then without a waterproof light source things are going to be pretty dark in those underwater tunnels anyway.

Over time the tunnels have ropes attached to them so that people can pull themselves to where they need to be or even pulleys so that things can be transported.
They could have breathing helmets placed in various places for emergency use including for people diving from the sea surface to use to escape into the tunnel network. If the helmets are tethered with the open end down then the air should remain inside.

Such a civilisation could engage in piracy along the coast. Sneak onto ships at night, steal what they need then dive overboard with their loot, escaping by diving down to the caves. The local merchants blame it on mermaids or somesuch, legends of people riding dolphins and then disappearing under the waves.

Similar legends exist on land but only rural people believe them, mysterious spirits that rise from the underworld to prey on the living. Most people guard their doors when such beings are spotted, none dare follow them back to the underworld.

Nope... Air is the biggest issue along with water tight buildings. Believe it or not, but water is very good at eroding away at materials and there would be no way to manually transfer enough air down underwater (its like trying to scuba dive but you have to go up and get air each time).

You would also have issues with the sand, as overtime salt will accumulate in the sand and you will need to source sand from somewhere where there is no salt which means going above ground or into fresh water rivers and transporting truck loads manually. It would also need to be isolated from the salt water until you are ready to use it as a filter.

Now disregarding the housing material you will use to build this (because I have no idea) I think the best way would to have some sort of special oxygen producing plant (like it produces visible bubbles worth of air). You could then build your houses as domes above these plants or integrate them into the walls of your plants to provide you with oxygen. (you would need something to remove excess air to make room, so many some special valve or tubing could be designed to accommodate this like in a toilet)

I would also adjust the biology of your people so that they can actually drink salt water. This would solve the huge issue of gathering and storing fresh water while you are already underwater.