When we study analysis on the real numbers (or complex numbers), we study the completion of the rationals via a natural way of measuring distance. There are other natural ways of measuring distance and they lead to other completions. The analysis there is strange from our accustomed point of view, sometimes it is easier, sometimes harder. The most important thing is to put all these spaces together and study the resulting object. It really is fundamental and it appears in many places. A deep study of this object and the objects derived from it is just in the beginning stages nowadays compared to the well-oiled machine we study in courses in real and complex analysis, ODE, PDE, differential geometry, etc.