Are there complications associated with a common cold?

Yes. A common cold triggers mild inflamation and drainage thru affected areas of the head & respiratory tract. Most infants and todlers will do fine and the symptoms pass. The extra drainage provides a bounty of extra food for germs that often barely get by in the body before being swallowed & digested. Extra germ growth may lead to ear, sinus, lung & throat infections producing a change in symptoms.
Yes. A common cold is another way of saying a 'mild virus'. Viral illnesses of course can come with a number of complications, most of which have to do with the medical history, age, and luck of your child. The younger the child, the more potential for a virus to cause more inflammation and trouble breathing. Ear infections, pneumonias and sinus infections are known issues. Most however, will do fine.