Euler wrote a piece on this titled "De la Controverse entre Mrs Leibniz
et Bernoulli sur les Logarithmes des Nombres Negatifs et Imaginaires",
Euler proves that each nonzero real number x has an infinity of logarithms. If x is positive then all but one of the logarithms are imaginary, if x is negative then all the logarithms are imaginary. He does this by use of the equation log(x) = nx^(1/n)-n, which is true when n is infinitely large, and familiar properties of the number of roots x^(1/n) and their properties for finite n. But i dont know about trig with imaginaries. I do no you can map time with Is tho with some 'interesing effects' acording to mr Hawking.