Looking for interesting unanswered questions within complexity theory for a project I'm looking for interesting open questions in complexity theory that someone with an undergraduate degree in math and comp/sci could theoretically tackle. I have strong interest in the polynomial hierarchy, and the study of probabilistic classes like RP, co-RP, ZPP, BPP, and also their logarithmic counterparts. I also have some interest in quantum computing complexity theory, but I'm afraid my knowledge of quantum isn't that strong. (Only the basics from my modern physics class, which included some quantum). Important: must be new research or new expansion on previous research, a question that either hasn't been asked or hasn't been answered successfully that I might be able to say something interesting about, even if I don't ultimately solve it. I'm willing to learn any necessary programming languages/programs/methods/etc to the end of answering this question (or trying to)! 

Note: this is for a senior year undergraduate thesis in math / comp sci. 

Thanks for any suggestions!

Edit: An example might be to analyze the complexity of a particular problem that hasn't been (formally) analyzed yet but may yield interesting results.
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.