What type of numbers are roots of $x^{2} = -1$ themselves? Are the roots $i, -i$ themselves irrational numbers or complex numbers or left auxiliary so that undefined? 

Just $i$ and $-i$. The fundamental theorem of algebra tells us that an equation like $x^2 + 1 = 0$ has two roots. In this particular case, $i$ is the principal square root of $-1$, and $-i$ is the other root.
The idea that $i$ could be irrational seems ridiculous to me, but I have to admit that I have never stopped to think why I would consider one complex number rational and another one irrational. The thing about the purely imaginary numbers is that in some ways they are just like the purely real numbers, only the axis of the former is perpendicular to the axis of the latter.
Let me ask you this: do you consider the number $-21$ to be rational or irrational? What about $-\sqrt{7}$? It seems to me that if you consider $-21$ to be a rational number, you should also consider $21i$ and $-21i$ to be rational, and if you consider $-\sqrt{7}$ irrational, you should also consider $\sqrt{-7}$ and $-\sqrt{-7}$ irrational. (And I'm using the square root symbol to mean the "principal" square root, but contrarians will still find a reason to howl, and to push the limits of serial downvoting).
But I would still stop short of redefining the symbol $\mathbb{Q}$; I recommend you only use that to mean purely real rational numbers.
