Example 3 If the equation $z^{m}=1+\mathrm{i}(m \in \mathbf{N})$ has exactly 2 roots whose corresponding points in the complex plane fall precisely in the first quadrant, then the value of $m$ is ( ).
(A) $3,4,5,6$
(B) $4,5,6,7$
(C) $5,6,7,8$
(D) $6,7,8,9$
Explanation: From the square root operation of complex numbers, we easily obtain
$$
\begin{array}{l}
\omega_{k}=\sqrt[2 m]{2}\left(\cos \frac{\frac{\pi}{4}+2 k \pi}{m}+\mathrm{i} \sin \frac{\frac{\pi}{4}+2 k \pi}{m}\right) \\
(k=0,1,2, \cdots, m-1) .
\end{array}
$$

Since the equation has and only has 2 roots in the first quadrant, we only need $\arg \omega_{1}<\frac{\pi}{2}, \arg \omega_{2} \geqslant \frac{\pi}{2}$, thus, we can get $4.5<m \leqslant 8.5$.
Since $m \in \mathbf{N}$, then $m=5,6,7,8$.
Therefore, the answer is (C).