A complex number is not a solution to a previously unsolvable equation. It's not a formal addition to our mathematical system, without meaning but with numerous application. It's not a square root of a negative number.
A complex number is a point.
And, of course, a complex number is all those other things too. But the entire story makes the most sense if a complex number is just a two-part number, a point in the plane. The fact that it can be all those other things is just a wonderful surprise.