On page 191 of the book "Einstein Gravity in a Nutshell," author Anthony Zee quotes Hermann Minkowski's stance on special relativity by saying, "The essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 3*10^5 km = square root of -1 seconds." Very, very interesting stuff relating time and space by a connection to imaginary numbers.
And in Tristan Needham's book Visual Complex Analysis, page 241 relates the complex plane to celestial mechanics. It gives the equation for an ellipse centered at the origin of the complex plane to be z=p*e^(it)+q*e^(it). Okay, that's fine. But what's REALLY surprising is if you square that whole equation on both sides. Translating z to z^2 you get (p*e^(it)+q*e^(it))^2 = (p^2)*e^(i2t)+(q^2)*e^(i2t)+2pq. The first two terms give you another origin centered ellipse, but the third term, 2pq, shifts the ellipse so that the origin point is now at the focus. So we have an ellipse in the complex plane with the focus at the origin point. Very much like Kepler's 1st law of planetary motion. But why should anything in the complex plane bear any resemblance to planetary motion? It's absolutely bizarre.
I believe there's a lot more underlying the structure of imaginary numbers that we have not yet found out. That's why they possess such a mystic property for me for the time being. When you first see that square root of negative one, you dismiss it as nonsense, but when you look deeper and see great and beautiful things coming out of them, it doesn't seem so much like nonsense anymore, even though you can't fully explain what's going on. It reminds me of God, who is easy to dismiss at first, until you look a bit more deeply and see beautiful and mysterious things that are difficult to explain.
