Is it worth buying carbon fiber road bike for short commutes? I'm planning to use a bike for rather short commutes (max half an hour), will the carbon fiber be noticeably more comfortable than aluminium one?
Comfort isn't really an issue for a 30-minute ride, as long as the bike fits. Also, for any given price point, an aluminium bike will have better components than a carbon bike, so will probably be more enjoyable to ride.

For a commuter bike, I'd be much more worried about theft and damage, unless you have somewhere secure to park the bike at work.  If you are worried that aluminium will be too harsh, you're probably wrong but I'd consider steel instead, rather than carbon. Or just put wider tyres on it. I've ridden rigid aluminium hybrids with 28mm tyres and had no comfort issues at all.

Any good bike shop should be happy to let you take a test ride of at least half an hour on any bike you're thinking of buying.