Which is better, a one handed War axe with shield or the two handed Battle axe?

Well depends on if you want buy the shields perks or not ^^.
Option one: is the super CC turtle tanker.
Option 2: is the larger damage and can still block & bash (block less effective and no elemental or arrow protection).
Option 3: Is take one handed buy the duel wield perks and use 2 War axes and chop every thing into little bits <- gives the highest damage of all 3 but no block/bash at all. Though you can switch to 1 + shield when it gets hairy.

^ Pretty much this. Kind of a silly question, which is better, neither is "better". Depends - do you like getting shot with arrows? Or would you prefer to crush things with only a few thunderous two-handed power attacks?

I personally mix it up a lot. Sometimes I use a two-hander, others dual-one-handers, and yet more other times a one-hander and a shield. I like variety

I like the feel of combat with the shield and axe or elven swoard.. The parry, the bash, the protection from magic and arrows.. It is fun to see a baddie crumble after two hits of a double hand weapon... but I like the mixup and timing of a one hand and the option to use magic in the other, then close then go back to close combat with a shield for the finish.

i use a 2 handed axe and i love it. the power and damage is so much more than that of just a one hander and i am still able to do a reasonable amount of blocking. you just have to make shure your stamana is high or you will be killed while your trying to swing your axe.

I'm loving the shield perks! Not long now before i unlock the charge! Deadly bash is good too!

Bashing them with the shield then cutting off there head is very satisfying Shield and sword/Axe is what i like best.