Quote from truefire;314544:
I'd say go magic. The beautiful thing about alchemy is that it has amazing synergy with the other three paths, at times almost as much synergy as magic has, and at times having even more synergy than magic. But the most important thing to note is that outside of maybe barrier spikes, alchemy has the least synergy with archery. Magic and alchemy can both be wielded while wearing a shield, and you effectively convert both bars into offensive powerhouses. You gain the support of healing others (Healing) or the power to heal yourself and punish the enemy at the same time (Life Drain). You even get to become a pseudo-melee specialist while still keeping your distance (Summon Golem), which also solves your problem of an alternative form of damage since magic and alchemy often run into magical active and passive defenses. You don't even need to equip your wand to bring out the devastating effects of your adv. spells; just put them into a bottle and you can chuck them while wielding a cylinder (Mana Crystallization) or even set and forget them when it comes to Ice Spear (ice mines). And, lest we forget, archery, which is a stam hog all on its own, would effectively increase your sp pot consumption which will conflict immensely with your crystal storage. While magic does the same thing with mp potions, it's important to note that you can switch between mana usage or stam usage on the fly. But the biggest competitor for bag space will be... arrows. A quiver of 1000 will hold you over if you don't plan to shoot often, but if you do, then that's a commodity you'd have to tack on in addition to more stam potions.
I could imagine an archer dabbling in alchemy to fill some gaps, but an alchemist can fill their own gaps better with magic or melee before archery.
(Crash Shot + Shock) - Easily one of the most deadly skill combinations in the game.
(Magnum Shot + Frozen Blast) - Archers have outstanding DPS. This allows you to hold an opponent while you pound into their lowered defense/protection.
(Barrier Spikes) - Obvious usefulness.
Alchemy has some usefulness combined with ranged combat. That being said, those skill combinations are super end-game.