I'm completely at a loss as to which one will be more important to my pure mage? Maybe I should just stick with all the magic schools?

I haven't played a mage beyond level 9. But I would say Alchemy would most likely be better because then you can make restore magicka/health potions and other things that let you conserve magicka for your more offensive magic. I don't see enchanting being that useful because most of the good mage apparel I find (like Archmages robes) already has better enchantments on it than the player could make.

it can be argued both ways. enchanting is pretty simple, provides constant boosts; alchemy is more tedious, provides a different kind of boost to each school than enchanting does, such as boosting the magnitude/duration of spells instead of decreasing the cost of them.

alchemy helps a TON.

enchanting is REQUIRED.

Required? How so?

free casting. i dont think it's required if you still want to be challenged in later levels instead of spammity-spam-spam-expert-level-destruction-spell.
*edit* twin enchanting is extremely useful, though not required. and i forgot no it's not required for free casting but does help in getting it.

Enchanting has always felt more "mage" to me than Alchemy, and that's what I base my decisions off of.

Nope but Alchemy Helps alot but you can just buy potions

true, but with few exceptions you can make much better pots than you can buy. the exceptions that come to mind are the ultimate pots of health/stamina/magicka.