The card text of "Hidden Ninja" says:
Special: Before a base scores, play a minion there.
Chronomage says:
You may play an extra action this turn.
Do I have the possiblity to play an arbitrary action after that?
The card text of "Hidden Ninja" says:
Special: Before a base scores, play a minion there.
Chronomage says:
You may play an extra action this turn.
Do I have the possiblity to play an arbitrary action after that?
In the "Void Where Prohibited" section of the rules it says "If you get to play extra minions or actions while a base is being scored, you must play those extra cards immediately or not at all" which means that you DO get to play the extra card from the Chronomage, but only if you play it immediately. You can't wait to see if other cards get played first.
While it may seem weird to be able to play actions during other people's turns, this ruling has been confirmed by an AEG rep here.
No, you cannot, at least not with the Chronomage. The card says "an extra action", but you didn't get any actions this turn (especially considering it wasn't yours).
Had it been your turn, however, and playing one of your minions would have caused that base to score, then the answer is yes.
I'm unaware if any expansions have added exceptions to this.