TITLE: Is it possible to construct a field larger than the complex numbers?
QUESTION [13 upvotes]: Can we extend the complex numbers in any way such that $\mathbb{C} \subset\mathbb{C}[a]$ ? Or is $\mathbb{C}$ the extension to end all extensions?

REPLY [11 votes]: Yet another form: ultraproducts. 
The cartesian product of fields
$$P = {\Bbb C}\times{\Bbb C}\times\cdots$$
isn't a field because has zero divisors:
$$(0,1,0,1,\cdots)(1,0,1,0\cdots)=(0,0,0,0,\cdots).$$
But a quotient will be a field. Let be $\mathcal U$a nonprincipal ultrafilter on $\Bbb N$. Define
$$(a_1,a_2,\cdots)\sim(b_1,b_2,\cdots)$$
when
$$\{n\in\Bbb N\,\vert\, a_n=b_n\}\in\mathcal U.$$
The quotient $F = P/\sim$ is a field strictly bigger (in the sense of cardinality) than $\Bbb C$
 and the inclusion $\Bbb C\longrightarrow F$ is...