Would it be difficult to show, in some way, what AI is currently moving (during the AI turn) like was done in Civ IV?

Can't you just select "SHOW ENEMY MOVES" and "SHOW FRIENDLY MOVES" from the options screen? Or are you asking for something else?

I'm referring to the old Civ II system where it would show a colored square in the bottom, right corner to indicate which AI was currently processing it's turn.
From a psychological standpoint, it is better to have a visual indicator like this to know what is going on and how much longer you have to wait when turns may take a long time to process. Of course, it doesn't tell you how long it will takem, but it gives you an idea of the progress, and you can guess about some of it from previous turns.