Advice needed re Carbon Fiber vs Shocks
There's no frame material in the world that will absorb bumps like a bike with a suspension fork/frame. 

However, the original "suspension" invention that made bicycles practical was the pneumatic tire. If you have a rigid frame and fork, that's all the real suspension you have. 

Skinny road tires at high psi have about 2-3mm of "travel", a fat commuter tire at 60 psi will have close to 25mm of travel. A suspension fork generally has about 100mm of travel. 

A CX bike is a good choice to build a commuter around since they generally take fatter tires ( usually up to 32mm ), but there's absolutely no reason to get a carbon CX bike for commuting. Especially one that does not take a rack or have fender mounts. 

With fat tires at the correct PSI, I think you would have a very hard time distinguishing between different frame materials in a blind test. Geometry and tire size make so much more of a difference than what the frame is made of. 

To enjoy your commute, I'd recommend finding a balance in the middle. A bike with suspension is going to be heavy and slow and overkill 99% of the time. A "racy" carbon CX bike is going to be fast, but rough and the handling will be a bit twitchy. A good commuter should take fat tires(32mm at least), have a relatively long wheelbase and slacker angles. 

But you can commute on anything. You might just try putting slicks on the bike you already have and see how that works for you. If you are going to get a new bike, I'd really suggest looking at CX/commuter bikes with disk brakes.