I wouldn’t get too hung up on “real” and “imaginary”. A complex number is just a mathematical object. You can equally well say that there are no complex numbers in quantum mechanics at all … only ordered pair of real numbers like (x,y). You can “add” two pairs by (x,y)+(z,w) = (x+z,y+w), and you can “multiply” two pairs by (x,y) × (z,w) = (xz-yw, xw+yz). (See here.) To answer your question more directly, you can multiply all states simultaneously by an arbitrary phase factor like −1 or i or -i and it makes no difference—global phase factors are unobservable. Sometimes people say that quantum states are complex rays, or points in a complex projective space, to make that point more clear. That actually makes the math more complicated and annoying in some cases … I learned this lesson by getting it wrong when writing the original version of this wikipedia article.
I wouldn’t get too hung up on “real” and “imaginary”. A complex number is just a mathematical object. You can equally well say that there are no complex numbers in quantum mechanics at all … only ordered pair of real numbers like (x,y). You can “add” two pairs by (x,y)+(z,w) = (x+z,y+w), and you can “multiply” two pairs by (x,y) × (z,w) = (xz-yw, xw+yz). (See here.) To answer your question more directly, you can multiply all states simultaneously by an arbitrary phase factor like −1 or i or -i and it makes no difference—global phase factors are unobservable. Sometimes people say that quantum states are complex rays, or points in a complex projective space, to make that point more clear. That actually makes the math more complicated and annoying in some cases … I learned this lesson by getting it wrong when writing the original version of this wikipedia article.