Well the article does mention:

If beauty doesn't motivate you sufficiently, then take solace in the fact that the properties of some differential equations which arise in physics and engineering are determined by polynomials you can build out of them, and the systems modelled by these differential equations are stable if the real part of all the roots are negative. Google for control theory to see lots more.

and

This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits.

and ... did you read it all?