We think of oceans and large lakes as being blue.
How can I get other colours in a way that is reasonably compatible with what we know from terrestial bio-geo-chemistry?
The easiest way to get a non-green/blue ocean is perhaps to use a life form. Just make sure your ocean has the ideal nutrients for them.
Salt Lake in Turkey Turns Red Because of Algae Bloom
Ammonia has a tendency to dissolve alkaline metals. This answer claims this would make the oceans very blue, this page suggests brown. The argument for blue is that solvated electrons make solutions deep blue; when the concentration is high ($[H]>3$M) the colour goes copper, which might be the source for the brown. Now, that is a pretty concentrated solution, so I would expect blue unless there were other substances adding colour.
Liquid nitrogen, helium and hydrogen are pretty colourless, but liquid oxygen is sky-blue (and under pressure and low temperature forms a lot of colourful crystal phases). Ozone is darker blue. Hydrogen perioxide is apparently a watery transparent blue. Of course, a planet with liquid oxidant oceans may have some habitability issues. I suspect that the same dispersion effects that make water blue will make liquid nitrogen, helium and hydrogen oceans look blueish.
Hydrocarbon have plenty of opportunities to acquire colour. Liquid methane is again colorless, and might look very clear on worlds like Titan where atmospheric methane has already filtered out the wavelengths that tend to be absorbed. However, the atmospheric haze consisting of photochemical smog is clearly yellowish and likely soluble in the methane. I would expect most hydrocarbon ocean planets to have oceans coloured by more complex tholin molecules, that would tend towards the red-yellow side. In extreme cases, maybe black.
Sulphuric acid is again colorless, but does tend to turn yellowish when contaminated with iron ions. And again organic matter and other substances can make it yellow, red, or black.
Water isn't blue. Or not very. Water appears blue-ish because most of the light you are seeing is reflected sky. It's not the same blue, because some of it is absorbed instead, so you are in effect getting dimmed skylight, or light blue mixed with black.
[ Here the surface is calm enough that there is only small amounts of surface scattering.
At the top of the picture you see a mottle of blue and dark bands. The blue bands are the back side of the wave. Reverse the light path from your eye, the water is at a shallower angle, so reflection dominates. The front side of the wave is show more light from the interior of the water.
Look at the reflection of the bow of the canoe. This area has a poor view of skylight, so the colour here is a mix of some bottom algae, and red light reflected from the canoe.
On a smoky overcast day, the water looks like hammered metal.
Water can be coloured by what's in it. Suspended clay gives it a brown colour. Dissolved tannic acid (peat bogs) along with micron sized peat particles make it the colour of strong tea in your cup, black if more than a foot deep.
*Smooth water in foreground and background shows bluish tinge of sky. White water is light scattered off of entrained air. The light brown is organics from the peat bogs that is the source for the water.
With algae in it water is green. There can be a substantial difference in green depending on which algae. Some algae are red -- hence red tide.
Runoff water from a glacier is often a milky gray initially. As it gets far enough and slow enough for larger rock flour particles to settle, it takes on a blue cast. The particles that remain are are close in size to the wave length of blue light, so it's scattered more.
Several comments have mentioned Crater Lake. On a day of broken overcast it doesn't appear particularly blue. I suspect much of it's reputation is due to being at high elevation, so the sky also is an extraordinary blue.
So now you have several mechanisms to colour water:
There is no reason for any of these to not work with other liquids.