that allows us to interpret complex numbers naturally as two-dimensional vectors in Argand diagrams, and the geometrical nature becomes more transparent. Both the real part and imaginary part of a complex variable can represent the real-valued physical coordinate of a one-dimensional system.
When performing differentiation and integration, it is easier to deal with exponential functions for complex numbers rather than trigonometric functions for real numbers. Also, instead of using the addition and subtraction formulas for trigonometric functions, we just use familiar rules related to exponential functions. Multiplication by a unit complex number "phase factor" is just a rotation in the complex plane. For a generic complex number, we can write in polar form \(z=r\ e^{i\theta}\), where \(r\) and \(\theta\) are both real-valued. Multiplication by a unit complex number gives \(\left(r\ e^{i\theta}\right)\ e^{i\phi}=r\ e^{i\left(\theta+\phi\right)}\).

4.2 Let \(z_1,\ z_2\) be complex numbers such that \(x_1,\ x_2\) are the real parts. Consider the original equations of motion in real-variables to be equations for complex-variables, and assume these equations are solved by \(z_1\ =\ Ae^{i\omega t}\)and  \(z_2\ =\ Be^{i\omega t}\). Note that we assume \(A,\ B\) are constants, but we allow them to be complex-valued (and are not simply assumed to be real-valued). So there could be different constant phase factors for \(z_1\) and \(z_2\). With the form of the solutions as assumed, plug into the differential equations obtained in 4.1, which then become algebraic equations since taking a time derivative is equivalent to multiplication by \(i\omega\).