If you rotate your phone's keypad 90 degrees, you can call Hogwarts!
To actually answer your question, complex numbers make signal processing algorithms much more tractable, and also make it easy to visualize many commonplace quantities and phenomena in electrical engineering. Anytime a value has both a magnitude and a time- or phase-based aspect, chances are you can use complex numbers to represent it.
Put another way: just as negative numbers allow you to specify movement along an axis, complex numbers allow you to specify rotation around an axis. Both are obviously pretty fundamental.
Not exactly. Complex numbers greatly simplify math in that they allow us to represent phase and amplitude, for instance, using only a single term. Usually this appears as e raised to some power of i (or j) that is multiplied by an angle. x and y coordinates can't really accomplish the same thing. If I had an (x, y) pair, y would tell me something about amplitude, but I wouldn't really know anything about phase without comparing x to something else.
The complex numbers form a ring that is isomorphic to the cross product of the ring of real numbers and itself....
C≅R×R
For those who haven't done abstract algebra:
This means your can map a complex number to (x,y), that every complex number maps to a unique (x,y), and that if you add or multiply two complex numbers and then convert the result to (x,y), you can convert the two complex numbers to (x,y) and then multiply getting the same answer. (it is also bijective, meaning it works both ways).
Importance: Complex numbers can be dealt with as coordinates and coordinates can be dealt with as complex numbers.
That's not true: the multiplication is different. (x1,y1) × (x2,y2) would have to be (x1x2 - y1y2, x1y2 + y1x2), not (x1y1, x2y2).
The addition does work as you suggest, so a more correct formal statement would be: the complex numbers form an additive group that is isomorphic to the cartesian product of the additive group of real numbers and itself; the complex numbers also happen to form a ring (indeed a field).
To actually answer your question, complex numbers make signal processing algorithms much more tractable, and also make it easy to visualize many commonplace quantities and phenomena in electrical engineering. Anytime a value has both a magnitude and a time- or phase-based aspect, chances are you can use complex numbers to represent it.
Put another way: just as negative numbers allow you to specify movement along an axis, complex numbers allow you to specify rotation around an axis. Both are obviously pretty fundamental.