What is really the purpose of $i$? We started talking about imaginary numbers again this year and asked this question in class, but nobody could really give a straight answer. So if anyone could tell me the real reason we have imaginary numbers that would be great! :)

The real reason? That sounds like you want a one-liner that you might not have heard before.
Take out the graph paper.
What happens if you keep applying $i$ to $1$? i.e, $\;i \times 1$, $\;i \times (i \times 1)$, $\;i \times (i \times (i \times 1))$, etc.
Apply $i$ to $1$ and you get $i$.
Apply $i$ to $i$ and you get $-1$.
Apply $i$ to $-1$ and you get $-i$.
Apply $i$ to $-i$ and you get $+1$.
So repeated application and you get to watch $1 \mapsto i \mapsto -1 \mapsto -i \mapsto 1 \dots$ 
Exercise: What happens if you keep applying $i$ to $1 + i$.

The real reason for $i$ is you get to rotate stuff by 90° using number
  multiplication.

There is nothing imaginary about rotations!
