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There's a human settlement at the north pole of the moon large enough to sustain expansion outwards for say 100 km in all directions. Humans like to think in terms of four principal cardinal directions: north, south, east and west. That works fine on Earth since since civilization has had no desire to live at the poles. But that won't be the case on the Moon -- the poles will be prime real-estate. There, notions of north and east will have little meaning with respect to local terrain. If someone asked you "which way is north?" the direction you would point to on the horizon would depend on your relationship to the pole at that particular time. Not a useful system for providing directions to lost travelers.
A more pragmatic system for direction bearing would be in relation to the Earth, since it would be near the horizon and generally in the same direction at all times (taking a small circular path slightly above to slightly below the horizon throughout the year). But what then would be good colloquial "homesteader" names for the four cardinal directions with respect to it?
Toward Earth might sensibly be called "earthward". But what about in the opposite direction and the two orthogonal directions?

9South. South. South. South. – Azuaron – 2017-12-14T21:58:26.760
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Humans currently live at the south pole. I don't think this is as much a problem as you think it is.
– sphennings – 2017-12-14T21:59:42.9671I saw a bear, so I stop my car and follow it on foot one kilometre east, then one kilometre north. I loose it from sight so I go one kilometre south and find my car at this point. What colour was the bear? – Legisey – 2017-12-14T22:02:14.963
@sphennings, then it should be easy for you to answer my question. Which you have not done. – dhinson919 – 2017-12-14T22:05:28.693
1They would use cardinal directions just like on earth Because it's what they know. – Vincent – 2017-12-14T22:18:42.640
If someone asked me where place x is in terms of the north, east, south and west directions and it's (say) 1 mile north, I tell them it's 1 mile north. Now, in a couple of days, let's say I'm a mile west of my current location and someone else asks for the same place in terms of the same directions, the relative location (because it's not an absolute thing) is now 1 mile east and 1 mile north. This is true any distance greater than a mile away from the exact north pole (although if defined well, can work even then) and anywhere not the exact south pole. So, why is it not a useful system? – Mithrandir24601 – 2017-12-14T22:19:38.363
A better (and more answerable question, that is less opinion based) question might be, What are logical options/reasons to name four cardinal directions from the poles So asking what environmental stuff could be a source for names. – James – 2017-12-14T22:21:34.050
Why not name your "Earthward" direction North then? And the others E W S? – Legisey – 2017-12-14T22:43:18.437
@Mithrandir24601, the normal cardinal directions would still be fine when working with maps. But let's say you're out in the field and you have radio contact with someone nearby and on the radio they say "Hey, what's that moving on the horizon over there?" And you say "Where?" What will be more likely to happen, A) the other person mentally calculates what longitude that are at so they can give you a proper map bearing, or B) they give you a direction that is relative to a fixed point that is almost always visible on the horizon, which would be Earth? – dhinson919 – 2017-12-14T22:49:37.503
@frouil, one might be tempted to refer to Earthward as, say, North. But then that would be confusing the term with the already established longitudinal direction. Depending where you are relative to the pole, Earth may be in the North or it may be South. – dhinson919 – 2017-12-14T22:51:57.740
Personally, I'd be happy figuring out relative displacements in order to find the actual position, but that's me, so I'm willing to accept that others view this very differently :P – Mithrandir24601 – 2017-12-14T23:07:09.870
@Mithrandir24601, :) you'd make a great astronaut. But I suspect the average homesteader would not be that mentally agile. – dhinson919 – 2017-12-14T23:10:50.733
@frouil The bear is the same colour as any other polar bear -- black. Anyone who says "white" is in error. – a4android – 2017-12-15T01:29:39.487
Maps and GPS would probably display locations using the MGRS grid system. Bearings and distances can be calculated or plotted between points on the MGRS topographical map. – Thucydides – 2017-12-15T02:20:34.180