Not sure of the origin of 'go south', but it has a general meaning of to decline, to move in an undesirable direction, to deteriorate. I can only assume that it comes from the fact that if you look at the compass, north is generally considered 'up' and south is 'down'. As for 'go west'. This was an exhortation in America during the 1800 and possibly 1900s when 'the west' was largely undeveloped by European settlers and probably relates to the gold rush too. If someone 'went west' they had the opportunity to get land, make their fortune etc. I think both these terms are becoming anachronisms and not in common use anymore - at least I rarely hear them. Perhaps Americans still use them.

Thank you for the helpful comment, Blair! The usage of "north" that you said is really interesting!

person = disappear company = shut(close) down operations.

It's probably an Amercian thing. You're right, it means something non physical is declining or is deteriorating. South could be used to describe the fall in person's health, a marriage, or a company's share price for example. I have heard 'north' used to say that a number is higher than it was previously. "Police say that the crowd was about 100,000. It was easily north of that." It wouldn't 'go north' it would 'be north'. East and west are not used as far as I know. It would be hard to picture what you mean if you used 'east' or 'west' to describe something non physical moving in a sideways direction. I'm not American and these are American English terms so someone else might answer this one better than me.