There is an argument to be made that the real numbers, by which I mean the completed reals, does not belong in an algebra course. In the answer above, all of the motivation comes from the real-algebraic-closure of $\mathbb{Q}$ (the largest algebraic field extension not containing $\sqrt{-1}$).
The reason one might want to introduce the whole continuum comes from numbers like $e$ and $\pi$, which are transcendental (the fact that these numbers are transcendental is not immediate and requires a proof that I would consider past middle-school level). If you're willing to state those facts without proof, you can give a moral argument for $e$ by showing that it is the limit of the sequence $((1+1/n)^n)_{n\in \mathbb{N}}$, which is Cauchy, and its inclusion in the real numbers follows from the completeness of $\mathbb{R}$. However, this argument may still be somewhat sophisticated for a middle-school algebra course.
Edit: On Prof. Clark's suggestion, I've copied my comments into the body text (with minor additions):
The notion of a sequence converging to a limiting value has a very intuitive geometric interpretation, so it wouldn't be hard to give a geometric argument (say on a graph, for example) that $e$ is a real number, since after relatively few iterations, the graph does level out. Showing that it is not the solution to a polynomial is effectively proving that it is transcendental, and I can't think of an informal argument showing this, but since this answer is community wiki, if someone has an idea, this would give a "moral" argument for the study of the "whole" continuum.