During analysis, how do you treat missing values?
good article on how to treat missing values
This might not be the most popular answer, but one option would be to use a method that does not require complete data to begin with, and which therefore obviates the need for missing data imputation.
For example, many probabilistic or generative modeling approaches are perfectly comfortable with some of the observations being missing.
Which particular probabilistic model should you use? Well, as a wise man once said, that depends on the problem.