If $E$ is a real Banach lattice, then complexification of $E$ is defined as follows: $$E_c= E+iE=\{x+iy:x,y \in E\}\\(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2)\\(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_2y_1+x_1y_2)\\|x+iy|=\sup_{\theta \in [0,2\pi)}|x\cos\theta+y\sin\theta|$$
What is the motivation behind describing $|x+iy|$ as above?
For real numbers $x,y$, then there is a $\theta\in[0,2\pi)$ such that $|x+iy|=e^{i\theta}\cdot(x,y)=x\cos\theta+y\sin\theta$ as the usual dot product in the plane.