Of course, the rules of cricket would demand that he inform the batsman each time he switches bowling hands. All mystery, and the element of surprise is lost. No wonder there are very few ambidextrous bowlers in cricket. Hanif Mohammed could bowl finger spin with both hands, and a few others are said to have bowled with their wrong hand at a point, generally as a bit of fun, not as a serious endeavour.
But it could be turned into a serious, if niche, endeavour if the law no longer required bowlers to inform the umpire in change of action. Bowlers could switch their action at will to adjust if the strike changes from a left-hander to a right-hander or just confuse bowlers who never know what angle the ball is going to come from and out of which hand as the bowler bounds to the crease. Spin bowlers could gain the most advantage from this, turning the ball both ways, using the same skill-set but different hands.
Ambidexterity is a skill rewarded in most other sports. Footballers who are genuinely as good with either foot are highly prized, rugby players are expected to be able to throw to either side, and there are no rules (as far as I know) preventing golfers driving in a right-handed stance before putting left-handed. Baseball players who can bat either handed are highly prized, and pitchers who pitch either handed rare but not prohibited.
Yet cricket promotes the inequality of batsmen being able to reverse-sweep and switch-hit, without bowlers able to counter with an innovation of their own. It's not like batsmen need more of an advantage in the modern game of big bats with edges as large as middles and postage stamp grounds with 60 yard boundaries.
Innovation has long been frowned on in cricket, when the googly was first invented it was derided and claimed to be morally wrong, tricking the batsman in such a way. There are palpitations amongst many about the doosra and its legality, despite the fact that several bowlers have been show to be able to bowl it without chucking.
It's time for cricket to embrace the ambidextrous, add another dimension to bowling and reward those who are, and those who can make themselves ambidextrous. Give it ten years and maybe the next generation of mystery spinners might come along and take the opportunity literally with both hands.