I think maybe combining elements should be more of an alchemist's job. I think for an elementalist, their strength should be adaptablity, switching between different tactics quickly and effectively. Here's my idea: while all four elements can be used for attacks, they also have different kinds of effects: water heals creatures and erodes armor, air is used to push things around or do things like allowing a creature to jump really high or temporarily run faster, earth is used for defending oneself from attacks and certain effects, and fire is all about all-out attacking.
That's what i think the elementalist should be about. As for the alchemist, he or she should be all about his special potions and experiments. The alchemist should have a deep understanding about what things are fundamentally made of. So that probably means training in arcane, like the wizard, but unlike the wizard the alchemist would be focused less on preventing his opponent's spells as changing their objects to his/her advantage, which would sometimes include destroying or inhibiting those objects in some way. The alchemist would be able to change the alchemical composition of an opponent's nonliving creatures in order to take control of and/or destroy them, reverse engineer some of their conjurations, i.e. making their mana flowers direct their extra channeling at him/her rather than their original caster, change the kinds of attacks that a creature is weak or resistant to, or even engineer an explosion, etc.
Of course knowing what your own objects are made of is important too. Building off of what Drealin said earlier, I'm thinking the bucket over the door trick could be used for a lot more than poison and such. It could be used as a way to give the alchemist's own creatures buffs and special effects and the like. Perhaps even effects that could change the alchemical composition of enemy objects upon contact. Also, I think there should be different chemicals that are more or less poisonous to different kinds of creatures. What's hazardous to a human mage might not be so much to a living metal creature, or a lava-dripping demon. I think that the whole poison thing sounds great for the alchemist, though. However, I think poison should be based on alchemical weaknesses. A creature would be poisoned when it consumes, is injected with or sometimes touches a chemical completely incompatible with its body. So if an alchemist throws a bag of water vapor into a zone where a fire creature is and the fire creature steps on the bag and breaks it, that creature as well as any other fire creatures in that zone would be poisoned, since water is anathema to them. That sort of thing.