I've taught a few people about the complex numbers, by stepping through expanding the naturals with the introduction of negatives to make integers, fractions to make rationals, irrationals to make reals, and finally (the 'novel' stage for my audience) imaginary numbers to make the complex numbers.

I emphasise the point that the new system always seems weird and confusing at first to the people who aren't used to it, and sometimes gets given a nasty name in contrast to the nice name of the old system (especially 'imaginary' vs 'real' and 'irrational' vs 'rational') but the new numbers are never more or less worthwhile than the old system - they're just different, and useful in new ways.