Yeah I looked up the moon proposals and fundamentally they have to go to L1/L2 points and they are not stationary on the surface or have to pivot around the poles. A tether that was lunostationary is impossible since it would have to cut through the earth

The anchor point on the moon will be fixed. It's the counter weight at L1/L2 that will drift and will need some (small, according to Wikipedia) correction.

No. Stop and think about it. The moon rotates every 28 days with respect to L1/L2.

I try! :-D

Isn't the moon tidally locked and that's why we never see the "far side" of the moon?

And the lagrange points are between us and the moon?

If so, I don't understand why the anchor on the moon can't be fixed.

For what it's worth, I think you're right, although it's been a very long time since I did any mechanics. The tidal locking isn't completely perfect, so the elevator would probably have to be basically a spool of rope rather than an anchor; and the Moon's axis of rotation precesses every 18 years ish per https://en.wikipedia.org/wiki/Cassini%27s_laws, so again there would need to be quite a lot of flex in the elevator; but I don't see why it couldn't be done.