Broken Base: The turn to a more theatrical style away from straight hard rock starting on the School's Out album and continuing once Alice went solo.
The 1980-1983 "new wave" (and alcoholic) period and the 1986-1991 hair metal "comeback" period. The former was commercially unsuccessful with the exception of the "Clones" single, but fans of those albums feel they have some of the best songs he ever wrote. The latter group of albums, especially 1989's Trash, brought Alice back into the mainstream but are disliked in some circles for being very much products of their time and lacking the personality of his best albums, often considering The Last Temptation to be his true artistic comeback (possibly foreshadowed by "Wind-Up Toy").
Michael Bruce and Glen Buxton vs. Dick Wagner and Steve Hunter can get heated sometimes, although most fans recognize that they fit the different styles of music from the albums they played on.
Surprisingly, Alice's conversion to born-again Christianity and inclusion of Christian themes on his albums largely averts this. The Last Temptation, Brutal Planet, and Dragontown are often considered his best work since Go To Hell, Welcome to my Nightmare, or even Billion Dollar Babies by fans and critics who would normally shun Christian rock. It helps that the albums are stylistically very similar to his earlier works (Alice already went to Hell once, after all) and that the lyrics focus more on general morality that happens to be derived from Christianity instead of more general Bible-beating that some born-again artists engage in.
Crosses the Line Twice: The most deplorable bits in his show are usually the funniest as well. One longstanding concert skit is for Alice to beat up a young actress while singing a very tender ballad about domestic violence, then immediately follow it up by dancing with a life sized ragdoll and singing about necrophilia. Beating your wife to death is horrible, waltzing with her obviously phony corpse is hilarious.
Ensemble Dark Horse: Dennis Dunaway and Neal Smith, the bassist and drummer in the original band, are very respected by the unfortunately all-to-small group of people who focus on the group's actual music.
Da Da is a critical favorite despite the fact that none of its songs have ever been played live and that Alice doesn't even remember recording it.
Even fans who loathe the hair metal period tend to like "Wind-Up Toy" and "Poison".
Harsher in Hindsight: the "I can't go to school cause I ain't got a gun" line from "Lost In America".
The song "Wicked Young Man" has become more relevant in The New '10s due to Neo-Nazis becoming more active than they have ever been before.
He changed his name to Alice Cooper because it was the best way to retain legal rights to its usage. Thus, the music company that the band worked for wouldn't be able to hold on to the rights. For contrast, see "the artist formerly known as Prince," who after switching companies became legally unable to use his original stage name.
This actually turned out to be unnecessary because he bought the band name, as well. Which means he was also able to trademark "Alice Cooper," which he would not have been able to do if it solely referred to him. Band names can be trademarked, but a person's name can't.
No Such Thing as Bad Publicity: Alice Cooper was the ultimate "kids love him, parents despise him" artist of his day, so obviously he just wallowed in this trope. Parents en masse raised an uproar through much of his career, and declared his music a danger to kids and teens, who promptly ignored that rhetoric and made Alice an icon.
During a 1969 concert in Toronto, Cooper picked up a chicken someone had thrown onstage and tossed it back into the audience, thinking since the bird had wings, it would fly. Instead, it plummeted into the audience, who promptly tore it to shreds. This story quickly mutated into a rumor that he'd actually bitten the head off the chicken himself. Cooper received a phone call from Frank Zappa asking if he'd done so. When he replied that he hadn't, Zappa said, "Well, don't tell anybody you didn't do it!"