Is this chord still a minor triad if you use a 4th in the bass or does the chord change? Can anyone think of any examples in pop or rock music where this is used?
I see three distinct possibilities based on how you are actually thinking about the chord itself:
If you really just consider it an A minor chord, it might just end up being a slash chord with a non harmonic tone in the bass. So it would be Am/D.
I would most likely name it D7sus2 which is spelled
D E A C. This makes most sense to me since there's no third, but a there is a 2nd/9th present which make sense for the sus.
Another possibility is if you want to look at all the notes together as one unit then I'd view this as an inverted C6/9 chord which is spelled
C E (G) A D. The G is not there, but omitting the 5th is quite common in actually playing chords which will show more of the quartal/quintal nature of the notes themselves, but is a streach naming wise.
I agree 100% with what Dom said, I just wanted to add that identifying a chords name often requires looking at its siblings and/or the key the piece is written in. Taking your example with A- and D in the bass I would interpret it as either
D E (F#) A C
D E (F) A C
because of the D in the bass. Not applicable to rootless voicings of course.