Let's call a morphism of schemes a strong immersion if it is an open immersion followed by a closed immersion. This is no standard terminology. The following facts are well-known (see Stacks project, 19.24.3, 22.2.8, 22.2.9, 22.2.10):

• Every strong immersion is an immersion.
• Every quasicompact immersion is a strong immersion.
• Every immersion with a reduced domain is a strong immersion.
• There are immersions which are not strong.

Now my question is the following: Let $X$ be an arbitrary scheme. Is the diagonal morphism $\Delta_X : X \to X \times X$ a strong immersion?

According to the facts above, this is true when $X$ is reduced or when $X$ is quasi-separated. One of the main difficulties with such questions is that we cannot work locally (for example, the scheme-theoretic image of $\Delta_X$ might not be a local construction), so that standard methods don't work. Nevertheless, I think it is a quite interesting question.