We have \begin{equation} \begin{split} i^2 = i i & = (0, 1) (0, 1) \\ & = (0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) \\ & = (-1, 0) \\ & = -1 \end{split} \end{equation} $\square$