
Red Metal |
Construct Rider archetype grants a construct animal companion.
Homunculist archetype grants a familiar.
Promethean Alchemist archetype grants a homunculus companion.
Tinkerer Alchemist archetype grants a clockwork spy familiar.

avr |

Various archetypes have a companion of some sort - homunculist, construct rider, promethean alchemist, tinkerer, possibly others.
With the spell knowledge discovery an alchemist can qualify for feats requiring casting, including craft construct (& its prerequisite feats).
The nature soul & animal ally feats can give anyone an animal companion. Similarly iron will/familiar bond (/improved familiar bond) can give anyone a familiar.

UnArcaneElection |

If for some reason you wanted an non-alchemical Familiar on an Alchemist, you could take the Iron Will - Familiar Bond - Improved Familiar Bond feat chain.
If for some reason you wanted a non-alchemical Animal Companion on an Alchemist, you could take the Nature Soul - Animal Ally - Boon Companion feat chain.
Edit: Ninja'd!
Some Eldritch Heritage feat chains and some VMC chains will get you one or the other.
These are all really feat-intensive, though, so generally they are pretty bad, but the right choice of VMC Wizard (gives full-progression Familiar at 3rd level) could be pretty good, depending upon what Arcane School abilities your particular build would find useful (several good possibilities, if you can afford the feats).

Derklord |

Variant Multiclassing: Druid provides a companion at level -4 at 7th level, which becomes a full level one at 11th level.

Derek Dalton |
Winged Maruader is a Goblin only archtype for the Alchemist but you get a choice between Dire Bat or Giant Vulture. It goes up like an animal companion. You lose Mutagen for it but you can always get the discovery for it later. Both are pretty sweet all things considered but I like the Dire Bat better.
Now this is up to your GM but mine and I have allowed Hobgoblins being a goblin subtype to pick Goblin feats and discoveries and archtypes. So go Hobgoblin +2 Dex and Con no minus. They with Alchemist favored class a bomb every 2 levels. The Undersize mount allows them to ride both Dire Bat and Giant vulture. There is a feat that allows you to take four levels of something else but your animal companion goes up. I'd suggest fighter to get combat feats for three of those levels.
There is a discovery that also gives you an Alchemical Zombie. Rules for it are outlined in the discovery. Not a perfect solution but it gives you something else to consider.
Personal note. Most arch types take away too much to be really useful in my opinion. There are about three or four I have seen and played that make them worth taking but overall I'd stick with the base class in most cases. The Winged Marauder is one that actually gives you something for what it takes away. It also gives you the option of actually getting Mutagen back. Half the Alchemist Archtypes I've read don't do that.

_Ozy_ |
Simulacra dont seem too useful though.. Takes 24 hours to grow, Costs 100 gold per hit die, And lasts for less then a day.. Might go for a animal companion through Animal Ally. Alchemist seems like a fun class, And if my Cleric dies the Alchemist could provide healing potions to aid the party!
The lesser simulacrum version is crap. The Greater Alchemical (i.e. normal) Simulacrum is amazingly useful.
It's 5x cheaper than the Wizard version and lasts forever. The healing rules don't make a lot of sense, though, since it's cheaper and as fast just to make an entirely new simulacrum than to repair a heavily damaged one.

VRMH |

my mind jumped to an alchemist picking up some stray dog and pumping it full of extacts.
You know, that could work as an archetype. Basically it'd be a mix between the Alchemist and the Summoner: an Alchemist with a mutated, steroid-drugged monstrosity.
A bit like the bizarro-counterpart of a Druid.