As a former alchemist, this got me to think about certain potentials.
See, some players cry foul at Destruction, claiming it is underpowered and that it can only be made useful by getting 100% cost reduction or close to it, or spamming Impact, both of which make it more a chore than fun to use. Others, like myself, assert that a mage needs a varied arsenal of spells and schools in order for Destruction to make a difference, and point out that it's not like Archery (not point-and-click) and instead has some defensive and tactical value, like in the case of walls and runes. But nowhere have I seen an argument that favors Destruction and Alchemy, and after some thought, I believe that these two skills synergize wonderfully well and have the potential to make Destruction one of, if not the, greatest source of damage in the game.
To begin with, Alchemy can restore magicka, increase your magicka pool and buff regeneration. Already, this can mitigate higher Destruction costs. To add to that, Alchemy is the only source in the game that can actually increase Destruction's damage output. Feel free to correct me, but the weakest potion I've seen available for purchase that increases the damage of Destruction spells is 40% for 30 seconds. That's a huge buff for a long time, IMO, and it only gets better: given use of weapons or Pickpocket, one can poison almost any enemy (other than undead, that is) with weakness to [element] poisons, weakness to all magic, and weakness to poison itself (and yes, these stack). Again, the weakest for-purchase versions of any of the poisons that I've seen are 40% weakness for 30 seconds, an enormous debuff.
Given the above, competency in Destruction and Alchemy can give us spectacular potential.
Fireball is an Adept spell that we're all probably familiar with, and with two ranks of Augmented Flames (because we are, after all, concerned with damage output and skill competency) does 60 points of damage in 15 feet, plus whatever DOT it does. Here's what that spell looks like when dualcast after drinking the weakest "+Destruction damage" potion (+40% damage with Destruction spells):
60 x 2.2 = 144 --> 144 x 1.4 = 201.6
With just that one dualcasted Fireball, we're already doing significantly more damage than the equivalent Master spell. If instead we had our Fireball spell on our offhand because we were whacking enemies with our mace and decided to poison them, we could have 40% weakness to magic, 40% weakness to fire, and 40% extra Destruction damage, amounting to a whole 120% extra damage output from a fire spell--in other words, the equivalent of dualcast damage with just one hand.
Of course, none of this is considering "weakness to poison" poisons and enemies who already have elemental weaknesses, or stronger, custom-tailored potions and poisons, for that matter

----------
Here are a few build ideas that work to support a character who uses Destruction and Alchemy primarily. Obviously, we don't care much for Restoration in these builds, given that Alchemy is a perfectly viable alternative for survival. Race is always up to the player, and I don't care to put up my preferred races for the builds (because I would probably say Khajiit every time).
Stealth Mage: Illusion, Pickpocket, Alteration
There are a lot of powerful advantages to stealth-centric mage builds, foremost among them (given Alchemy and Destruction, of course) is that the guilds who would favor such a character are overflowing with alchemical ingredients, even at the beginning of their questlines.
With Illusion and Destruction, we won't have any need of Sneak. There are no sneak attacks with magic, and Illusion provides a means of subterfuge, as well as Quiet Casting (and other nifty tools, assuming you don't make invisibility potions and don't have muffled equipment).
Pickpocket offers a whole lot of options to stealth-based characters in the first place, such as premeditated defenses and a means of income, but for a Stealth Mage (who doesn't have any weapon skills), it also means we have a way to poison NPCs--and a way to follow up poison with a quick dualcast!
Alteration, while seemingly out of place, offers protection, tactical value and more subterfuge. Obviously, there are the armor spells, but beyond that, we have a silent(!) Paralyze (which works well with Pickpocket and any form of damage dealing; almost game-breaking), silent life-detection, silent Water Breathing for escapes, and even silent Telekinesis for long-ranged larceny or throwing bear traps and sweet rolls at people. And there's nothing wrong with silent armor spells when preparing an ambush, either.
Battlemage: One Handed, Heavy Armor, Enchanting
Battlemages are always a force to be reckoned with, and this one is no different. Weapons provide a means of poison use, armor for protection, and Enchanting...for everything.
When a build makes use of Magicka, Health and Stamina, Enchanting is a must-have skill; throwing buffs to attributes on your armor is a lifesaver that helps to mitigate the fact that you're spreading yourself thin every level. On top of that, we have such wonders as "weakness to fire" poisons applied to flaming war axes, served with a side of Incinerate. Toasty!
----------
Thoughts? Criticisms? I want feedback of any kind; discuss!




You get some of these effects from different spells from different schools of magic, though.