Do there exist "O(1)-complete" problems?
Since reading the input is necessary for almost all problems, we need at least $\Omega(n)$ time for nearly all problems, where $n$ is the size of the input.
Therefore, you may think of the class of linear time problems, which is already defined.

However, we still do not know any $O(n)$-complete or $O(n^2)$-complete problem. The field of fine-grained complexity has some new results in this area, but the classes are problem-based (for example, APSP is equivalent to Radius, Negative Triangle, ...).