Do they stack?
This doesn’t stack with any similar increase from traits such as Magical Knack.
BM/MK are race and magical traits. Multidisciplined is a racial trait. It states that traits wouldn't stack, but does it mean the "traits" traits, or are the racial traits included in this? The "such as" phrasing being a linguistic undecidability is once again an issue here. It could be such as this trait that increases caster levels under given circumstances or it could be such as this trait (as in a trait you can pick from http://www.d20pfsrd.com/traits/ ).
Alchemists treat their Class level as caster level, could they benefit from such an enhancement in caster level?
Could it be worth it foregoing multitalented and consequently multitalented mastery if you're going 10 levels into a prestige class? 10 hit- or skill points (possibly 3 more, respectively FCB) at the cost of 1 feat and two (or possibly more) caster levels.
Multidisciplined and Magical Knack will stack, up to your character level, since they are totally different properties that happen to share the word "trait."
I am not familiar with Bifurcated Magic and cannot find it on AoN or d20pfsrd. Could you link to it?
Alchemists don't actually have a caster level, so they will not benefit. They just use their class level as an effective caster level in various places.
No opinion on the trade-off you're considering.
http://archivesofnethys.com/TraitDisplay.aspx?ItemName=Bifurcated%20Magic
You’ve studied the methods of both elven and human spellcasters. Pick two classes when you gain this trait—your caster level in each of those classes is treated as 1 higher as long as this doesn’t raise your caster level above your total Hit Dice. This doesn’t stack with any similar increase from traits such as Magical Knack.
The writing of Bifurcated Magic seems to work with Alchemists handling of "caster levels".
For Alchemist it says
The alchemist uses his level as the caster level to determine any effect based on caster level.
Let's say you're wizard 1/alchemist 1, you would prepare a expeditious retreat potion and according to bifurcated magic you'd treat the effect of the potion as 1 caster level higher, namely 2.
BM/MK are race and magical traits. Multidisciplined is a racial trait. It states that traits wouldn't stack, but does it mean the "traits" traits, or are the racial traits included in this?
Using the word 'trait' for both groups of character options is unfortunate, but hard to fix nowadays, I guess.
The relevant thing is the type of bonus:
Multidisciplined gives a racial bonus to CL.
Magical Knack provides a trait bonus to CL.
Bifurcated Magic also adds a trait bonus to CL.
Bonuses of different sources stack, so Multidisciplined can be combined with both other options. Magical Knack and Bifurcated Magic don't stack, but together you achieve a +1 and a +2 bonus to CL, because the higher bonus counts. Or a +1 to a first class, +1 to a second and +2 to a third, if you want to go really crazy with casting classes.
The phrase "This doesn’t stack with any similar increase from traits such as Magical Knack." doesn't introduce a new restriction, just remembers the player of a general rule (bonuses of same type don't stack).
Alchemists treat their Class level as caster level, could they benefit from such an enhancement in caster level?
Hmm, I'd rule in favor of the alchemist here, but actually there is a FAQ stating that alchemist doesn't qualify for most crafting feats - likely because of the missing caster level.
Could it be worth it foregoing multitalented and consequently multitalented mastery if you're going 10 levels into a prestige class? 10 hit- or skill points (possibly 3 more, respectively FCB) at the cost of 1 feat and two (or possibly more) caster levels.
Multitalented Mastery is useful if you go crazy with combining normal classes, but otherwise it does much less than Toughness or Cunning. It's an option to add it on top of these two feats, if you don't mind the diminishing returns (less HP / ranks per feat).
The Favored Prestige Class feat might be interesting for you.
@Sheepish Eidolon
Now I must think up a way to get all these traits to work together in a build that isn't completely gimped.
That's quite the tall order though.
The favored prestige class sounds much better than multitalented mastery.