The book is wrong.
It should say:
where n is an integer.
When n = 0 you get one of the real fourth roots. This is the one given in the book.
There is obviously another real fourth root, as you rightly point out. It's given by n = 2.
Then there are two other fourth roots, both non-real, given by n = -1 and n = 1. These are the ones that bobak makes mention of.
Four fourth roots in total, as expected.
Other values of n just give you the same four roots again.