In a lot of fix bed applications you want the support to essentially do nothing other than be hard and keep the product from moving. Some critical chemical applications will use the high purity, alpha alumina, supports because they have little effect on process streams.
There are applications where an "active support" can be quite helpful as you can use some of the support volume to aid in whatever the reactor is supposed to do. Activated aluminas (tabular) are often used as support for desicant applications and for support in HF absorbents because the high surface area is good for being able to pick up contaminent. You probably don't want to use them for supports in a chloride absorbent because of the potential for undesirable side chain reactions.
I would think that an activated carbon support could have some nice properties as a purification absorbent support.