I initially said that quantum mechanics is generally not considered a complete theory, but several people in the comments disagree, so I guess I'll have to make a more explicit case for this. In any case, it seems this matter is not as "settled" as I thought it was. If you just want my examples of questions, just skip to the bulleted points.
Specifically, the following unanswered question is a central one of modern day theoretical physics:
- What is the best way to unify quantum mechanics and gravity?
Notice that I said gravity, not special relativity. Quantum field theories are effective in unifying quantum mechanics with special relativity, but they don't touch gravity. Indeed, when you apply quantum mechanics to regimes where both quantum effects and relativistic effects are expected to be important, e.g. the area around black holes, quantum mechanics gives us answers that are wrong. Perhaps we have different definitions of complete, but I'm not sure how you can say that a theory that makes wrong predictions is complete.
Some people were also arguing that this example lies outside the domain of quantum mechanics. I'd argue against this notion. Quantum mechanics, while only noticeable at very small scales, is expected to (and largely does) reduce to the classical results at larger scales. Quantum mechanics does not e.g. predict that people will randomly pass through walls (with any reasonable change), it only predicts that particles will. Similarly, Special relativity and General relativity correctly reduce to our observations of slow-moving objects and less-extreme gravity. So, I think it is imprecise to argue that certain regimes lie outside quantum mechanics in any sense that allows quantum mechanics to be straight-up wrong in those domains. I think that a complete theory would not be so clearly inconsistent with reality. Again, perhaps we are thinking of different notions of complete.
Another example of a question that has not been answered:
- What is the correct interpretation (e.g. Copenhagen, Many-Worlds, etc...) of quantum mechanics? This question might never be answered, and indeed might not be a meaningful question, but it is also possible that it is a meaningful question, and might be answered in the future by new data that takes these ideas farther from meta-physics. In any case, while we certainly don't have an answer to this question, I did not mean to imply that the status of this question implies the incompleteness of quantum mechanics.
These are very "high level" questions, and perhaps you were looking for something more concrete. Some examples of more concrete questions:
- Find an analytic 3-body (i.e. wavefunctions of the two electrons and the nucleus) solution to the helium atom including the coulomb potential and and electron-electron repulsion.
- Develop an accurate theory of high-temperature superconductivity.
I might be mistaken, but I'm pretty sure none of the questions above have accepted answers. A couple examples of computationally difficult problems that do have solutions:
- Solve the hydrogen atom potential for an electron in parabolic-cylindrical coordinates.
- Find the entropy of a system of neutral atoms trapped in an optical lattice.
Hope this helps