Mapping the Territory

This activity follows on from Complex Puzzle.

Use the Geogebra interactivity below to find some pairs of complex numbers whose product is a real number. What do you notice?
Can you explain it algebraically?

Use the Geogebra interactivity to find some pairs of complex numbers whose product is an imaginary number. What do you notice?
Can you explain it algebraically?

In general, what would you need to multiply by $a+bi$ to get a real number?  Or to get an imaginary number?

For a given complex number $a + bi$, what would you need to multiply by to get to another given number $x + yi$?
How does this relate to your geometric interpretation of multiplication of complex numbers?