"Imaginary" and "real" numbers are so-called because at one time, it was thought that real numbers exist and imaginary ones don't. We now know that this is not true, but the labels "real" and "imaginary" stuck around, which is unfortunate because they are misleading.

Imaginary and real numbers are both real in the sense that you can do maths with them. And they're both imaginary in the sense that they're concepts, not physical things.

It's also worth noting that imaginary numbers are not just a contrivance for taking square roots of negative numbers, they are a core part of mathematics with uses far too numerous to list.

The one who isn't walking on a great circle (i.e. an "equator" of the sphere) is constantly turning away from you, in the sense that if they imagine their path laid out in front of them, it appears to curve to one side.

Put another way, imagine doing this not with walking but with cars with the steering fixed dead ahead. Try it with toy cars and a basketball if you have those objects handy.