Add and multiply perplexing numbers
Python 2, 62 bytes

def f(s):b,a=[eval(s)/2.for j in-1,1];print'%d+j*%d'%(a+b,a-b)

We simply evaluate the expression s with  j=1 and j=-1, and output half their sum and half their difference as the coefficients of 1 and j.
This works because both j=1 and j=-1 satisfy the defining equation defining equation j*j==1. So, the original and simplified expressions must be equal for both these values. The simplified expression is linear, so this gives two linear equations in two unknowns:
x + 1*y  = s(1)  = 2*a
x - 1*y  = s(-1) = 2*b

which is solved by x=a+b, y=a-b.