Non-trivial example of algebraically closed fields

Another concrete example is given by Puiseux's theorem:

If $K$ is an algebraically closed field of characteristic $0$, the field $K\langle\langle X\rangle\rangle$ of Puiseux's series is an algebraic closure of the field of formal power series $K((X))$.

Note:

$K\langle\langle X\rangle\rangle=\displaystyle\bigcup_{n\ge1}K((X^{1/n}))$

In On Numbers and Games, Conway defined a field structure on the set of all ordinals, and he calls the result $\mathbf{On}_2$. It is an algebraically closed field of characteristic two, if you are willing to ignore the fact that it's really too big to be a set.

It is also possible to "cut" $\mathbf{On}_2$, that is, to only consider ordinals smaller than a given limit and to get some algebraically closed fields. For example, the ordinals smaller than $\omega^{\omega^\omega}$ give the algebraic closure of $\mathbb F_2$, cf. this Lenstra's article.

Here's an introduction to this construction.

You can start with $\Bbb Q$ and take its algebraic closure $\bar{\Bbb Q}\subsetneq\Bbb C$ and you get an algebraically closed subfield of $\Bbb C$ that's much much smaller than $\Bbb C$ (countable versus uncountable). Then you can add any transcendental to it like $\pi$ and you can take the algebraic closure of that $\overline{\bar{\Bbb Q}(\pi)}$. So you can produce infinitely many algebraically closed subsets of $\Bbb C$ in this way. What makes $\Bbb C$ special is not just that it's algebraically closed but that it's also complete.

Other examples are the p-adic fields which have complete and algebraically closed extensions which are very different from $\Bbb C$.