A good question to ask ourselves. I expected my answer to be straightforward, but I'm going back and forth with myself about it. :)
My username here is my actual name. My profile has my full name on it. I intentionally chose to represent myself as myself with this account. The goal is to be as authentic as I can so that the relationships I build here are based on who I truly am. Repelling is just as important as attracting and if people aren't digging me they can move along. I won't have to try to guess what to do to keep followers or engagement.
On the other hand, I've been working to build a brand that is not based on my name. I try to keep that more neutral and professional.
I feel like I can have both but it's best for me to keep them separate. That way I don't have to worry about my quirkiness contaminating my business branding and I have a channel for somewhat unfiltered self-expression.