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" (http://mathforum.org/johnandbetty/) takes such an approach.

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) takes such an approach.