[dragonwriter]

> a complex number is a point in the plane (in the same way as a real number is a point in a straight line)

More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line)

Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.)

But real numbers aren't points on a line, and complex numbers aren't points on a plane.