how do you factor y^4-1 completely

first, it is a difference of two squares:

(y^2-1)(y^2+1)
Next, the first term is again the difference of two squares. If you allow imaginary numbers, then the second term is a difference of two squares (y+i)(y-i)

you should recognize the difference of squares

y^4-1
= (y^2 + 1)(y^2 - 1) , ahh, once more
= (y^2 + 1)(y + 1)(y-1)