My son 15 years old has a two dark brown-purple spot under his big toe?

Probably bruising: I am assuming the spots appeared very recently. That is a reasonable place for bruising to occur from physical activity like running. The thickness of the skin on the bottom of the toe makes the color of a bruise different from bruises on other areas.

Spots: This can't be diagnosed with the information that is available. A history, physical examination & other studies may be needed to determine cause/causes.