Why algebraic closures?

Let me begin by summarizing the question:

Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations?

Historically the solution for polynomials was important, and people were trying to find a good way to test when a certain polynomial has a root. This led to talking about algebraically closed fields, where polynomials have roots. I will particularly focus on the rationals from now on.

How do we construct the rational numbers? We begin with $\{0,1\}$ and we say that if we add $1+1+\ldots+1$ we never have $0$. We begin by closing this set under addition, then under subtraction and then under division.

However we can consider the alternative, we iterate from $\{0,1\}$ and at every step we add solutions all four operations on the elements we have thus far, so the construction would go like:

1. $\{0,1\}$, we begin. Next we add additive inverse, a term for $1+1$ and multiplicative inverse for $1$:
2. $\{-1,0,1,2\}$. Now we add the additive inverse for $2$, the addition of $1+2$, and multiplicative inverse for $2$:
3. $\{-2,-1,0,\frac12,1,2,3\}$. Now we add the sums possible with the elements we have so far, the missing inverses, and so on:
4. $\{-3,-2,-1\frac12,-1,-\frac12,0,\frac14,\frac13,\frac12,1,1\frac12,2\frac12,3,3\frac12,4,5\}$. We continue ad infinitum.

It is not very hard to see that any rational number is in this set, and that this set forms a field (indeed the rationals).

Consider now the family of operations $\exp_q(x)=x^q$ defined for $x\geq 0$ and for a rational number $q$. If we reiterate the above algorithm from $\mathbb Q$ and close it under $\exp_q(x)$ for positive $q$ we end up with a subfield of the real-closure of $\mathbb Q$. The result, while not the entire real algebraic numbers, is radically closed. Any number in the field has a root of any rational order. If we only take closure under a limited collection $\exp_q$ functions we will get a subfield of this field (e.g. close only under $\exp_{0.5}$).

Consider now the Sine-closure of $\mathbb Q$:

1. $Q_0=\mathbb Q$, we begin with the rationals.
2. $Q_1=Q_0\cup\{\sin(x)\mid x\in Q_0\}$, the rationals were closed under field operations, so we only need to add $\sin$'s.
3. $Q_2$ is the collection of all sums and multiplication of pairs from $Q_1$, adding additive and multiplicative inverses, and adding $\sin(x)$ for $x\in Q_1$.
4. $Q_3$ constructed the same.

We finish by taking $\mathbb Q_{\sin}=\bigcup_{n=0}^\infty Q_n$. This is a field which extends $\mathbb Q$ and is closed under the function $\sin$. This field contains transcendental elements and is countable, so it is a non-algebraic subfield of $\mathbb R$.

So why are we mostly interested in algebraically closed fields, and not in fields like the Sine-closure of $\mathbb Q$ (or perhaps other construction similar to this one)?

Is the reason historical, is it because algebra and analysis are somewhat disjoint in their purposes and analysis takes $\mathbb{R,C}$ to begin with?

Nowadays we are interested in such number systems. Indeed, in any symbolic mathematics system that handles calculus (e.g. Macsyma, Maple, Mathematica), one is forced to be interested in general "elementary number" systems closed under "elementary" operations, e.g. see Chow's monthly article What is a closed form number? But such transcendental number theory is much more difficult than algebraic number theory. There are many open problems with no resolution in sight, e.g. Schanuel's Conjecture. There has been much interesting model-theoretic work done on such topics in the past few decades, e.g. see work by Daniel Richardson and Lou van den Dries to get an entry point into this literature, and see the literature on symbolic mathematical computation for algorithms and heuristics.

As far as I can tell, the first field you have constructed is not the real algebraic numbers; you only get real algebraic numbers whose minimal polynomial has solvable Galois group (and you don't even get all those, e.g. because of casus irreducibilis).

Anyway, dealing with transcendental numbers is hard! As far as I know, there is no algorithm for deciding equality of elements of your field (I don't know if this is open or known to be undecidable). Deciding equality of algebraic numbers, on the other hand, is straightforward: compute minimal polynomials, then compute the two numbers you're interested in to sufficiently high precision until their digits start to disagree.

There is also a lot one can say about algebraic numbers, and this is already good enough for many applications (e.g. algebraic closures suffice to give us eigenvalues of linear operators on finite-dimensional vector spaces). One can also talk about the algebraic closure very generally of any field, whereas operations like $\sin x$ are specific to characteristic $0$ and require the existence of a topology relative to which the defining series converges.

The natural objects appearing in mathematics closed under lots of operations like $\sin x$ are usually not fields (although model theorists are interested in exponential fields), but rather things like the smooth algebra $C^{\infty}(M)$ of smooth functions on a smooth manifold $M$ or Banach algebras. The keyword here is "functional calculus," as in holomorphic functional calculus.