Alumina is a standard choice as a polishing paste, because it's hard, cheap, chemically inert...
It won't dissolve into water, and shall not in order to polish. If it's fine enough, it will stay suspended in water for a long time, true.
I had powders of tungsten carbide and of graphite, both around 30nm size of individual particles (graphite makes bigger clumps). Very dilute in water, tungsten carbide takes days to settle partially and can be mixed again. Lighter graphite stays longer. More viscous liquids would keep them suspended longer.
Indian ink is a graphite suspension but finer, and stays for years with little settling.
Now, polishing requires a paste rather than a dilute suspension, doesn't it? I'd say settling won't happen then. But to get an optical finish, you need grains much smaller than a quarter of a 500nm wavelength, so <<100nm is the right direction, for instance 30nm.
Not fun at all: nanoparticles or TiO2 have been recently suspected of being unhealthy. And they're everywhere, including in toothpaste and sunscreen creams.