i is useful in all sorts of contexts, since when multiplying two complex numbers together you simply multiply the moduli, but crucially sum the arguments (angle of rotation). In other words, complex numbers are super useful for describing anything rotating.

'Imaginary' is an unfortunate phrase for the square root of minus one which I think was down to an insult from Descartes because he believed they were fictitious. It suggests that complex numbers are less 'real' than negative numbers - for this reason John Conway proposed calling them 'lateral numbers', which might help clear up some misconceptions.