Looking for interesting unanswered questions within complexity theory for a project
Do something interesting with visibly languages. They are a relatively recent topic with many useful practical applications. For example, I would find it interesting to better understand the role of synchronization here, compared to the role of synchronization for the theory of codes. Already the simplest case could be interesting, where the visibly language just distinguishes between inner symbols and separator symbols, and the considered codes are just bifix code.

David Richerby is right that this questions allows opinion biased answers. The origins of the opinions voiced in this answer can be found here.