As long as the liquid can carry enough oxygen to the lungs to remove enough carbon dioxide from it, we can, in theory, breathe it.
In the 1960s, scientists tried to experiment with this on mice. It had a degree of success, as the mice could breathe under the fluid, but would die shortly after because the fluid couldn’t remove carbon dioxide as well as it could carry oxygen.
Perfluocarbons are currently the closest thing we have to breathable liquids because they are can carry oxygen and carbon dioxide twice as well as air. When tested on mice, it was confirmed that the mice could go back to breathing gas afterward. Perfluocarbons are continually being refined to be more viable as breathable liquids.
Despite all the progress, there is still one major difficulty. Our lungs are meant to carry air, not liquid. It takes much longer and takes much more energy to transfer liquid in and out of our lungs than it does gas.
Our lungs aren’t strong enough to carry liquid, and so even though perfluocarbons are better at carrying oxygen and carbon dioxide, they are so much harder for our lungs to circulate that the benefit is lost. So for now, breathable liquids remain in the realm of fiction. But scientists continue to test on them, and one day may find a liquid solution that is of low enough density for our lungs to breathe it with ease.