Which one is correct? And why?
- "That would require that he imagine something."
- "That would require that he imagines something."
Both can be.
The former suggests that whatever-it-is would demand that he imagine something then and there. The latter suggests that there be a requirement that he already imagines something.
This is because the sentence structural position of "that he imagine(s) something" can take phrases of different nature, and a requirement can be to do something, or for some condition to be met. The first example is a required action, while the second is a required condition.
Even for the action, you would usually say "he imagines" - that's how it conjugates in the third person singular, after all. But when an action is stated to be required things work differently. I think, but I'm not sure, is that it uses the infinite, and sometimes infinitives can be bare - as they are when there is a requirement using "that". "I require you to be silent" is using a non-bare infinitive, while "I require that you be silent" uses the bare infinitive.