Sorry Bill G I forgot if you understood the above you should now get a Reimann sphere which will complete the answer.
What reimann did was say why does the solution space have to be a flat grid (mxm flat earth) why not make it a sphere (like earth). Infinity is now simply the point futherest from you which is the point that is antipodal to where you are now.
It's easy to see the logic if we look at earth if I travel on the surface in a direct line in any direction the point futherest from me is the point antipodal to where I am.
It gives you one correct answer no matter how large or small the sphere. The infinity of the point on earth from where you are now is the point on the opposite side of the earth from where you currently are if you can only travel on the surface. As per earth you must however use some form of polar co-ordinates on the scheme you can't use a square grid of complex numbers.
So there is one complex number system that I believe has a definite number infinity and many shaped solution space would do the same.
The achilles heel is you never proved that such a thing is justified and could or should exist, we just made it so based a familiar background. I growl at Bill S when he does that so I need to be consistent
So I am still happy to say I am not suggesting I am correct, but I can create some good jokes to the "authorities"
