Ask any mentally ill person if you need to be "Normal" if you want people to be kind to you and treat you respectfully and you'll get the answer yes pretty consistently. We know better than most the importance of fitting in. When you don't people ostracize you. Being a part of the group is how we get our needs met as human beings.
Look at people like Lady Gaga, she's very successful. But she's also very different.
A lot of the things we regard as normal would have, at one time or another, be considered very outlandish. Success is not dependent on conformity. It is independent from things like this and depends almost exclusively on hard-work and sometimes Reaction.
Because for every conformist, there are at least two other people who would rather Fly than Walk.
So it really depends on what you want to "succeed" at. That's up to you. Depending on what that is you may need to toe the line and conform on certain things or not. It depends on what success means to you. The way people talk about "success" and ram it down people's throats these days is oppressive. People have the right to define "success" in what ever way they want.