The Complex Conjugate Theorem states that, given the polynomial $p(x)$ with real coefficients $p \in \mathbb{R}[p]$ and one of its roots being $a+bi \in \mathbb{C}$, its complex-conjugate pair $\overline {z}$ must be a root as well. We could also intuitively conclude that only by multiplying $(a+bi)$ with $(a-bi)$ will the imaginary parts be destroyed, allowing $p$ to be a real polynomial.
Following this line of reasoning, it shouldn't be necessary for complex roots of a complex polynomial to come in pairs. However, this premise was used in solving one of my school assignments.
Do complex roots always have to come in pairs, regardless of the field in which the polynomial was defined?