Complex Values in Second Order Differential Equations

Recently I've learned that second order linear homogeneous differential equations can be solved by assuming the function to be something like this.
$$Ay''+By'+Cy=0$$
$$y = e^{St} $$
$$AS^2+BS+C=0$$
When encountering underdamped systems, the value of S would be imaginary, leaving you with Euler's identity.
$$e^{i\alpha}=\cos(\alpha t)+i\sin(\alpha t)$$

When solving for the fundamental solutions our professor disregarded the imaginary coefficient and claimed that the fundamental solutions are the imaginary component and the real component. How is this so??