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.