The polyester finish on your Alembic is similar to automotive paint but harder (which allows it to maintain a more mirror-like finish). You can use auto polishes to fix the problems; go to a automotive paint shop to get the best quality stuff.
To create a finish with a mirror gloss, you need to have a level and totally smooth surface at a microscopic level. When the finish was originally sprayed on the surface is pretty smooth but not particularly level. You start with a fairly coarse abrasive which cuts through the finish and levels the surface, but leaves big scratches. After the surface is level, you switch to progressively finer polishes which remove the scratches from the previous abrasive until it's smooth as a mirror.
In your case, handling has created scratches in the surface. You'll start with a fairly coarse abrasive and polish out those scratches. If you can feel roughness with your thumbnail going across the scratches, then you need something really rough like rubbing compound. If the surface is matte but feels smooth, then a white polishing compound may be sufficient. You want to use the least abrasive compound that will do the job, since each pass will remove some of the finish. If you use too fine of an abrasive, it'll take forever to get the deep scratches out and you'll never completely remove the marks. Take your bass to the auto paint shop and they can set you up with exactly the right abrasives (they probably have 4 grades of sandpaper that are smoother than glossy paper!).
A progression of 3-4 abrasives will probably get you completely back to the original appearance.
Although there's been some different advice here, you normally want to work with linear strokes when doing this kind of polishing. The reason for this is that it's hard to fully remove the deep scratches from the coarser abrasives if they're circular. Using a rough abrasive with a circular buffer is the cause of swirl marks on cars, which you don't want.
Black finishes are the hardest to maintain as they show imperfections the most.
David Fung