What is the motivation for quaternions? I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?

Complex numbers do indeed enable us to solve certain previously unsolvable condition equations; but that is not the motivation for them. It is unfortunately common to confuse application with motivation. 
Complex numbers extends the idea of quantity as algebraically-expressed-location to two dimensions after "negation" introduces it in one dimension.
The obvious next extension is to three dimensions; but alas, the same neat properties from one and two dimensions don't hold in three dimensions, as related by others above. That doesn't mean three dimensions is uninteresting; but just that this notational device for representation is less interesting. 
