When can a conjugate root be assumed?

One of the roots of the equation $z^4-z^3+4z^2+3z+p=0$ is $1-2i$, where $p$ is a constant.

(i) A student says that “One of the other roots must be $1+2i$.” Explain why this statement is not entirely correct.

(ii) Determine the value of $p$.

(iii) Find the exact values of all the other roots of the equation.