When you talk about improvising over a C chord, do you mean that someone else is strumming a C chord and you are improvising lead over that chord?
If 'yes' - then you can't just go changing to another chord when you feel like it. You must follow the rhythm guitarist's chords. If he or she changes to A minor then you should do the same, i.e., playing the notes of the A minor chord, (and other passing notes, etc.) like you did with the C chord.
If playing by yourself, improvising bits of chords and lead too - then you can change whenever you like. You are in full control. If you stick with chords that belong to the key, it will sound correct, normal and possibly boring. If you slip in some out of key chords it might sound terrible - but it just might sound amazing. That's the fascinating part - exploring 'the unknown'..
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