The students from St Stephens School, Australia, found the following examples:
• Pairs of complex numbers whose sum is a real number:
2+i, 2-i
4+5i, 3-5i
• Pairs of complex numbers whose sum is an imaginary number:
2i+1, -i-1
i+2, i-2
• Pairs of complex numbers whose product is a real number:
1+2i, 1-2i
2+3i, 4-6i
• Pairs of complex numbers whose product is an imaginary number:
3+3i, 3+3i
1, i

Sina Sanaizadeh from Hinde House Secondary School, Sheffield, sent in the following explanations:
• In general, what would you need to add to a+bi to get a real number?
We want to remove the imaginary part of a+bi to get a real number, so we want to add a  complex number of the form c-bi.
• Or an imaginary number?
We want to remove the real part of a+bi to get an imaginary number, so we want to add a complex number of the form -a+ci.
• In general, what would you need to multiply by a+bi to get a real number?
Consider the product of the complex numbers a+bi and c+di:
As a and b are fixed, must have $\frac{c}{d}$ = $\frac{-a}{b}$.
As a and b are fixed, must have $\frac{d}{c}$ = $\frac{a}{b}$.