As I assume you know, longer wavelengths such as red and orange are lost very quickly as depth increases. Therefore, regardless of the light capture capacity of the camera, which is very important (since a camera that captures color at 0.1 lux is not the same as one that captures color at 0.001 lux), it is undoubtedly essential to have good lighting, as explained in the following link.
Despite this, this can be corrected to a certain extent, if your camera has manual WB, from what I see in the video you share, you have to raise the red color much more, and not lower the green as much. On the other hand, lower the saturation a little to slightly eliminate the milky appearance of the water.
You have to play with these parameters as you increase the depth, but of course from a certain point without artificial lighting you have nothing to do. Since no matter how good the camera is, it cannot show a light wave that does not exist due to its loss due to depth.
This is a comparative example of a quality IP camera, with well-corrected manual WB, and the old BR2 camera in parallel without correction.
La prima cosa da fare é accendere i fari, io qui in Italia nel lago di Iseo, ho sempre acqua molto torbida, e senza la luce giusta diventa difficile operare. E se vuoi filmati con poca sospensione devi allontanare i fari dalla camera, in modo di avere una zona buia davanti per almeno 60cm. E lascia impostato il bilanciamento del bianco in automatico, dopo varie mie prove con tante camere, é ancora la soluzione migliore.