Yeah, the amplifiers need 10,000 volts. The way the cable is constructed is to have the fiber pairs inside a copper tube (there's some other stuff in between too). The voltage is carried by the tube. The ocean itself serves as the ground.
In general the power to the amplifiers is series connected and the ocean does not come into the question at all. If it did you would have huge issues with corrosion even ignoring the effects on the ocean itself. So, along the cable there is a wire that carries small DC current that goes between the amplifiers, each amplifier places zener diode along this wire and gets its power from it. At each landing station there is a current source (that is capable of developing significant open-circuit voltage) that powers this (this scheme is the reason why the cables are laid not only in loops, but in "loops-of-loops").