Given the structural formula of the active substance in red cabbage indicator. Is it possible to determine what colours it will absorb and reemit from first principles? I am guessing if it did it would involve alot of quantum mechanics, but surely it is possible to use the shrodinger equation to figure out all the allowed electron energy states ?

Yes, it is possible to calculate the absorption spectrum (and so much more) of the different species involved in red cabbage indicator from first principles. Density functional theory (DFT) calculations would probably be the way to go. It is entirely possible to optimize the structure of each of the relevant molecules, and at the same time calculate the energies and symmetries of the orbitals (occupied and virtual), which will directly determine the wavelengths of the photons it will interact with (and a rough estimate of the intensity of the absorption), which is exactly the info you would need.

The actual mathematics is pretty gnarly (lots of very large matrices being multiplied together, and subjected to operators, and eigenvectors being found), but there are computer programs that do all the crunching for you as long as you know what to ask them for.