> I'm teaching college algebra right now and introduce 'i' algebraically as a solution to x^2+1=0... but then we talk about graphing quadratics and there's no simple connection between the geometry/graph and the algebra.

Take an equation like x^2 -2x + c

When C is some large negative number, the roots are symmetric about x = 1. As you increase C, roots are real until C = 1, basically the two roots meet in the middle.

When you increase it beyond 1, the roots become complex numbers, but they stay symmetric, they "lift off" from being real into being complex, but still symmetric (conjugates) where 1 is always the real part but the imaginary parts become sqrt(C - 1). You can conveniently imagine the complex plane on top of the real XY plane and the roots go orthonogal to the direction they were going when they were real, from the point where they met.

That's kind of how I visualize it for quadratics. For higher orders you cut a complex circle into n equal pieces. Haven't quite figured it out in my head yet.