Why does my 10 month hands and feet turn purple and cold?

Acrocyanosis. Being cold causes vasoconstriction (tightening down) of the blood vessels in the periphery. This is called acrocyanosis. Hands and feet (and lips too) turning blue is usually from this and rarely a sign of heart disease, but you can check the tongue and inside of the mouth -- if these parts are pink when the hands/feet/lips are blue, then it is unlikely to be related to the heart.
Typical. This is a typical response in an infant to heat loss. Their temperature regulatory mechanism is immature and over corrects for body heat loss by constricting blood vessels in the extremities. If the baby is otherwise healthy and has had normal check-ups, rest easy.
Cold hand and feet. This is often associated to raynaud's disease. Have your chil checked by pediatrician.