Completion of Complex Numbers In some way, $\mathbb{C}$ completes $\mathbb{R}$, why is there nothing that completes $\mathbb{C}$? Is it just more so that we don't want anything more than $\mathbb{C}$, or is there a property of $\mathbb{C}$ that makes it complete, in some sense? I know algebraically $\mathbb{C}$ is very nice, but is there nothing else we look for?
 A: A structure $X$ has to be completed if some property or operation we'd like to have is only partially present in $X$. E.g., in ${\mathbb N}$ you can universally add, but only sometimes subtract. Therefore ${\mathbb N}$ is completed to ${\mathbb Z}$. In this larger structure subtraction is universally possible, and has the desired properties.
Similarly ${\mathbb R}$ is completed to ${\mathbb C}$, because we desire a working environment where the equation $x^2+1=0$ has a solution. Now it turns out (this is a miracle) that the smallest such enlargement, namely the set of all numbers of the form $x+iy$, with $x$, $y\in{\mathbb R}$ and $i^2=-1$, not only is a field, but contains the hoped for number of solutions to all polynomial equations whatsoever.
So if you wanted to complete ${\mathbb C}$ even further you would have to name a problem which somehow makes sense in ${\mathbb C}$ but cannot be universally solved in ${\mathbb C}$. Maybe you'd like to create a theory of "complex infinitesimals" and then would arrive at some (very large!) superstructure $\tilde{\mathbb C}$. 
The simplest example of a completion of ${\mathbb C}$ consists in extending ${\mathbb C}$  to the so-called Riemann sphere by adding the single point  $\infty$. This is a great idea in connection with Moebius transformations, or rational functions in general, and eliminates a lot of exception handling. But this particular completion of ${\mathbb C}$ is of no much help when we are dealing  with the function $t\mapsto e^{it}$ or similar  complex-valued functions occurring in mathematical physics.
