An Alchemist can obtain a Tumor Familiar. There's also the feat Improved Familiar, which is accompanied by a table which lists Arcane Spellcaster Level in one of the columns and has the prerequisite:
Prerequisites: Ability to acquire a new familiar, compatible alignment, sufficiently high level (see below).
Can the alchemist class fulfill the sufficiently high level prerequisite and take Improved Familiar?
Answering a really old question with an FAQ written after the question, Paizo has ruled that tumor familiars cannot become Improved Familiars:
Second, tumor familiars, as lumps of flesh in the shape of animals, can’t become Improved Familiars.
First, the alchemist would need a for-real familiar. The discovery tumor familiar very carefully never says that the tumor becomes an actual familiar:
The alchemist creates a Diminutive or Tiny tumor on his body, usually on his back or stomach. As a standard action, the alchemist can have the tumor detach itself from his body as a separate creature vaguely resembling a kind of animal suitable for a familiar (bat, cat, and so on) and move about as if it were an independent creature. The tumor can reattach itself to the alchemist as a standard action. The tumor has all the abilities of the animal it resembles (for example, a batlike tumor can fly) and familiar abilities based on the alchemist’s caster level (though some familiar abilities may be useless to an alchemist). The tumor acts as the alchemist’s familiar whether attached or separated (providing a skill bonus, the Alertness feat, and so on). When attached to the alchemist, the tumor has fast healing 5. An alchemist’s extracts and mutagens are considered spells for the purposes of familiar abilities like share spells and deliver touch spells. If a tumor familiar is lost or dies, it can be replaced 1 week later through a specialized procedure that costs 200 gp per alchemist level. The ritual takes 8 hours to complete.
(Emphasis mine.) So the tumor familiar's form is suitable for a familiar, the tumor familiar has the abilities of a familiar, and the tumor familiar acts as a familiar, but the tumor familiar is still technically not a familiar. (Yes, that's terrible.)
Second, although the alchemist can copy the arcane magical writings from a wizard's spellbook, the alchemist doesn't have an arcane caster level. The Alchemist on Alchemy says
Although the alchemist doesn't actually cast spells, he does have a formulae list that determines what extracts he can create. An alchemist can utilize spell-trigger items if the spell appears on his formulae list, but not spell-completion items (unless he uses Use Magic Device to do so). An extract is “cast” by drinking it, as if imbibing a potion—the effects of an extract exactly duplicate the spell upon which its formula is based, save that the spell always affects only the drinking alchemist. The alchemist uses his level as the caster level to determine any effect based on caster level.
(Emphasis mine.) Unfortunately, the alchemist level that the alchemist uses for his extracts is neither arcane nor divine.
:-)
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Commented
Aug 18, 2015 at 20:34
Rules as Written, No.
Improved Familiar's text specifically refers to the chart which specifics arcane spellcaster level. The alchemist isn't an arcane spell caster and doesn't have a 'virtual level' of such for the purposes of his familiar. Instead his familiar gains abilities based on his alchemist's caster level (again something he doesn't have technically).
Rules as Intended, Maybe.
According to Thomas Jacobs (Creative Director)
Alchemist caster levels equal their alchemist level. Note we don't generally call it a "SPELLcaster level." Neither do we call it an "ELIXIRcaster level." The rules apply for all spell or spell-similar effects.
There appears to be a little intent that an Alchemist... could. How game breaking would it be? Ask your GM. Me personally, I wouldn't have a problem with it, until you wanted a tumor larger than yourself.
