ShopBot will do 3D carving, and you can get a rotary indexer (think CNC lathe) addon for the 4th axis.
You're looking at roughly the ShopBot Desktop 24-18, don't see any reason you need a water cooled spindle, the router version is very good, I have the full size table with a router, and unless you are a full time production shop the spindle is an expensive upgrade (around $1200 more I believe on the Desktop) Add a 3" diameter indexer for $950 or a 6" Indexer (mas turn 8") for $2250. One of these days I'm gonna add one to my full size CNC, but most people have them and say they sit collecting dust, that they just don't see much need for one once the "fun" playing with phase wears off.
Call them up, and they'll invite you into the shop for tour and demo of the machines. Worth the trip.
And consider used, ShopBot supports them, and you can save $1000 to $2000 off new when they come available, just watch the ShopBot forums. I bought my full size used, 3 years old, and saved about 1/3 the price of new.
The new Vectric software supports two sided carving, so you can carve on both sides of a material. You design in the software, and carve one side, flip the material, and carve the second side.