Lines in the Complex Plane

What is the equation of a straight line in the complex plane? There are many different forms, but I want to look at some of the simplest ones. If you know the slope \(m \in \mathbb{R}\) and intercept \(b \in \mathbb{R}\) of the line, you can write an equation in parametric form \[ z = x + i (m x + b) ,\] where \(x \in \mathbb{R}\) and \(z \in \mathbb{C}\).