Then, I learned about trying local U(1) symmetry and trying to make the Langrangian invariant under that and wham, you get EM for free, without even postulating it a priori. I was amazed--all of a sudden, I realized the reason why all these particle physicists were so bent on postulating this or that symmetry, hitherto not seen, might exist in nature; I realized how fundamental symmetries were and how deeply intertwined with conservation laws they were.
http://en.wikipedia.org/wiki/Wick_rotation
and there was this on HN some time ago
https://news.ycombinator.com/item?id=8377680
Though I didn't find Scott Aaronson to be as clear as the professor who finally showed us the Wick rotation. Unfortunately he passed away a few years back right after having missed being awarded the Nobel prize by an inch.
However I think I really fell in love with it when I understood gauge invariance.
In general I think you can always try to tell your students things a little early to impress them, but it rarely works until they can work out the math themselves. I remember our professor in introductory quantum mechanics saying something along the lines of: "and now you see why quantum mechanics is just Markov chains in imaginary time", but until another professor showed us the Wick rotation in the context of path integrals nobody really appreciated that even if it could have been in our grasp earlier.
I guess this has come out way denser than I meant it to be, but my point is that you can always try to introduce something a little bit earlier, but you'll often find that your students don't want to learn some diluted crap, they want the real deal!