Ok, so I decided to look for advice on my build on the paizo forums. I'm going for a four-armed two-handed weapon dual wielder. There is no consensus and no specific ruling about this. Vestigial limbs state they can hold things (like a shield or a weapon) but don't add actions. With dual wielding it's still only two attacks, the limbs helping to wield two two-handed weapons. (four hands, two two-handers, right?) My thoughts (because I want to play it because it's wicked cool) looking at the entry for vestigial limbs is that it is legal as per RAW.
But considering this is E'n, not the Pathfinder Society, I would like to know if it is legal here. My entire character concept kind of depends on it so I thought it best to ask before I started levelling then found out I'm useless.
Legal, but, well, it is painful to try. Two-weapon fighting with anything other than light weapons is -4/-4 to hit. The BAB lost with alchemist is another -1. So, you are running at a -5/-5 to hit compared to an equivalent barbarian/fighter swinging once. You start stacking power attack penalties on that and he quickly becomes a flurry of misses. You can overcome most of those penalties with Rage + Mutagen, but rage time is counted in rounds. You are only good for one fight a day, maybe two.
It also puts you in the permanent freak show class. That has social consequences. LPF play is slow. It will take about a year before you get to fourth level where you would get the 4th arm.
But, if you do want to go down that path, a faster way is 2 levels of Ragechemist, 1 of Synthesist, then Barbarian for the rest. For the Synthesist, take biped, the Extra Limbs and Increased Natural Armor evolutions. That makes you a four handed beast at third level with +6 NA, Mage Armor, +2 Will over the other version, and you can drop the "freak show" attribute at any time and walk around town as a normal human. Overall for your career, you are at BAB=level-1 which is reasonable for triple classed. The synthesist form also gives you darkvision, which can be a major bonus at times. This path also allows you to rearrange the mental stats a bit, as the physical stats are locked in by the eidolon. Just keep CON at the same or higher than the eidolon so that you don't drop HP if you fall out of fused form. Switching one of the traits to Magical Knack(Synthesist) will allow you to cast the two spells you have at caster level=3. Those spells would be Mage Armor and Lesser Rejuvenate Eidolon. (aka, armor and healing) You can still be superstitious about any one else casting spells on you.
Even faster is the one level dip in Synthesist as your first level. Alec is just a tool for the eidolon to get access to fun drugs. Once it is here and established, it sends you down the alchemist path before going for pure natural rage.