• "Numbers'' are really abstract things, not just "quantities''. They can correspond to transformations --- for example, dilations and translations of the line. Show how this corresponds to multiplication and addition. Negative numbers are flips. This explains "negative times negative is positive," and shows $x^2=1$ has two solutions.
• Solving $x^2=-1$ corresponds to answering the question "What can you do twice to get a flip?" Likely as not someone will think of rotating 90 degrees. Dilations, translations and rotations of the plane are complex numbers.