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?
Best Answer
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:
(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
(Emphasis mine.) Unfortunately, the alchemist level that the alchemist uses for his extracts is neither arcane nor divine.