In my baseball dreams, I was always a pitching/hitting star, a sort of Babe Ruth who could switch hit and switch pitch. Because of platoon splits in baseball (left-handed hitters tend to struggle more against left-handed pitchers, and right-handed hitters tend to struggle more against right-handed pitchers, though the effect is not as pronounced as with left-handed hitters), being an ambidextrous pitcher would be an advantage assuming you actually were competent pitching from both sides.
Such a pitcher exists today, and he's pitching at Creighton University. Within the article, I realized that one theoretical advantage of such a pitcher is negated by the rule that "a pitcher must declare which arm he will use before throwing his first pitch and cannot change before the at-bat ends." Otherwise a pitcher could force an opposing manager to use up pinch-hitters by switching arms after they'd been announced.