I think in precalculus students should be taught the following:
- Euler's identity for $e^{i \theta}$.
- The principal value of $log(x)$ for $x<0$.
Then in Calculus they should be taught that
$\int dx/x = \log(x)+C$ instead of $\log|x|+C$.
Likewise, teach them that
$\int dx \tan(x) = -\log(\cos(x))+C$
and so on. I don't think that would be too advanced. The advantage would be, that, what they learn will be consistent with what some will learn at a later date in complex analysis. Perhaps more important, students would get the expected answer from tools such as Wolfram Alpha and Mathematica. Any thoughts?