According to The Walking Dead, the only functioning part of a zombie's brain is the brain stem, being the central nervous system, etc etc. No memories, thoughts, or otherwise. Only smell, and sight. I doubt they have taste, either.
So essentially, the zombie is not brain-dead, but it's organs are all but shut down. The virus in The Walking Dead focuses on keeping the brain stem, and heart alive, all other organs don't matter as much, the only other one is the heart.. That's why in the second or third episode, you see a zombie with no legs, stomach, and exposed ribs with most-likely punctured lungs moving and pulling itself toward the protagonist.
I conclude that they are only alive in the brain and heart, so technically alive, but not able to reason between what we call "right and wrong", they have no memories, dreams or thought. I'd say half dead.