Area as a complex number?

While we're integrating over a real variable, the integrand is complex. With the linearity condition $\int_a^b(f+ig)d\psi=\int_a^bfd\psi+i\int_a^bgd\psi$, the problem reduces to calculating the areas under two real-valued functions.

Consider a complex number instead as a real vector of length $2$ dotted with the vector $\langle 1, i \rangle$. For example, $3+5i = \langle 3, 5 \rangle \cdot \langle1, i\rangle$. Then a function $f(x) = f_r(x) + if_i(x)$ is equal to $\langle f_r(x), f_i(x) \rangle \cdot \langle1, i\rangle$.

The integral of $f(x)$ would then be equal to $$\langle1, i\rangle\cdot\int_a^b \langle f_r(x), f_i(x)\rangle dx = \langle1, i\rangle \cdot\left\langle \int_a^bf_r(x)dx, \int_a^bf_i(x)dx\right\rangle$$

This sort of reasoning could also extend to quaternions and other number systems. Each integral is the sum of the individual integrals (or areas) multiplied by the quantity ($1, i$, quaternion units, etc).