The solubility of oxygen in water just isn't high enough to make it practical to extract enough oxygen for breathing. You would need something like 50 gallons per minute just to satisfy your basic resting metabolism, which in turn would require a ton of power and/or surface area for gas exchange. (Multiply that by a factor of 3 or 4 if you intend to actually do any swimming.)

So you would have to bring a supply of oxygen gas with you, and at that point you might as well get it into your body the old-fashioned way instead of relying on a dangerous and fiddly ECMO machine.