Origin shift in the complex plane

Question

I'm just learning about basic complex geometry, and we were taught the locus for a complex line that passes through the origin, arg(z) = θ. But this form of a complex line is cleaner than the others, so I thought of generalizing this same equation for a line that DOESN'T pass through the origin. How can such an equation be derived using an origin shift in the Argand plane, to a fixed complex number z' = Re(z'), analogous to the Cartesian Plane, where the X intercept is non zero? Any other generalization way would be helpful too. Thanks in advance.