Remember that i is the name of the twizzle that has a number arrow equal to the unit arrow rotated through 90 degrees.

In other words i = 1 cis 90 .

The twizzle multiplication rule is neatly expressed by saying 'Multiply the number arrow lengths, and add the angles'.

So $i^2=i\times i=1\mathrm{cis}90\times 1\mathrm{cis}90=1\mathrm{cis}180=-1.$ Now you should have enough information to multiply out (z-i)(z+i) .