Is it too abstract to motivate complex numbers in terms of the equations we can solve depending on whether we choose to work in ${\mathbb N, \mathbb Z, \mathbb Q, \mathbb R, \mathbb C}$?
The famous "John and Betty" ([Link](https://web.archive.org/web/20180101080419/http://mathforum.org/mbower/johnandbetty)) takes such an approach.