To my knowledge (and it may even say this under the extra limb discovery) is that having extra limbs doesn't provide extra attacks. So with 3 arms you can wield 3 different weapons and choose to swing with any 2 of them using two-weapon fighting. I assume that the same would be if you were holding 2 bows, you don't get any extra attacks. It could be interesting flavor.
Remember that two-weapon fighting works best with light weapons, so even if it somehow works you would be stacking massive minuses to hit anyway.
Question back for you.. why 4 arms? 3 would allow you to hold two bows and use the 3rd arm to draw each one in turn as you attacked.