and are complex numbers.
is the product
Move the points and around to see what happens to

Fix , and move until is on the x-axis. What can you say about the trajectory of as you move it to keep on the x-axis?
Repeat the above for other values of :
Can you make predictions about where needs to be for to be on the x-axis?
Can you predict where needs to be when you want to be at a given point on the x-axis?
Can you use your understanding of multiplication of complex numbers to explain how to make these predictions? Take a look at A Brief Introduction to Complex Numbers for a reminder of the notation and algebraic manipulation.
Now carry out the same process but this time aiming to keep on the y-axis.