Can a Tumor Familiar become an Improved Familiar?

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?

Even with the discovery tumor familiar, an alchemist can't normally take the feat Improved Familiar

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.

• I feel that "[t]he tumor acts as the alchemist’s familiar" is sufficient for that. Since the prerequisite is "ability to acquire a new familiar", I'd like to know if there's any other instance where the wording ("If a tumor familiar is lost or dies") is similarly so slightly different (i.e. by the addition of one (adjectival) word) that would be undesirable if interpreted thusly. – NiteCyper Aug 18 '15 at 20:18
• @NiteCyper You mean like damage, nonlethal damage, and ability damage? Or like cover, partial cover, and total cover? Or like caster level and arcane caster level? I'll see what I can find. :-) – Hey I Can Chan Aug 18 '15 at 20:34
• Then I wonder for less official terms or terms that are potentially synonymous that I feel tumor familiar is, but your examples are good. Thank you. I mean, does nonlethal damage count for things that trigger upon taking damage? Aren't the latter terms of cover and caster level subsets of the former term? – NiteCyper Aug 18 '15 at 20:36
• @NiteCyper Maybe? Probably. I mean, there are always things people can argue about, anyway. Terms aren't necessarily inclusive—for example, look at all those different kinds of cover, but we'd probably agree that a longsword is a sword even though longsword is one word (unlike, for example, short sword). There's not a trading card game-like precision to Pathfinder's language, but more of a talk-to-the-GM-and-have-him-rule-on-it kind of expectation for handling ambiguity. – Hey I Can Chan Aug 19 '15 at 0:42
• I feel like the last emphasis is most important; it is clear that you do not 'have' a CL, and it is frequently cited that Alchemists, while magically inclined, are not Casters in the sense of most in-game functions; you can't metamagic extracts for example. From the PFSRD FAQ "As written, no, alchemists are not spellcasters, and therefore can't select feats such as Craft Wondrous Item... " – Ifusaso Aug 26 '16 at 20:47

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.

• As "for most intents and purposes", it is identical in function to a normal familiar, I, too, would allow taking the feat, though with proportional changes to the ritual length, and a restricted list of creatures to choose from - which would remain tumorous. – kaay Aug 18 '15 at 10:41
• @kaay What are you quoting by "for most intents and purposes"? – NiteCyper Aug 18 '15 at 20:30
• @nitecyper A house rule - to allow ingenious improvisation to substitute many checks. Like a mage allowed to take fighter feats but only use them (and those they're required for) if they own a tk charm of their own making. It's like the Physics of Discworld, or a Use Magical Device check on reality. Rarely imbalanced, and fun to play. – kaay Aug 19 '15 at 4:16