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}).