The noun for that kind of breathing: when someone is breathing too quickly or too deeply, taking in more air than the body needs, and feeling dizzy or strange because of it. Doctors talk about an attack or a spell of hyperventilation. People often meet it in panic or great fear, and sometimes after very hard exercise. The textbook picture is too much oxygen and dizziness. Though actually, as with the verb, you are also losing too much of the used air your body breathes out, and that is a large part of the dizzy, tingly feeling. The person may be sure they cannot breathe, even while they are breathing a lot. This is the name of the state; the verb is hyperventilate.