Here we present examples of totally real fields, totally imaginary fields and CM-fields.

Examples:

1. Let $K=\Rats(\sqrt{d})$ with $d$ a square-free positive integer. Then $$\Sigma_K=\{ \operatorname{Id}_K, \sigma\}$$ where $\operatorname{Id}_K\colon K \hookrightarrow \Complex $ is the identity map ($\operatorname{Id}_K(k)=k$ for all $k\in K$ , whereas $$\sigma\colon K \hookrightarrow \Complex,\quad \sigma(a+b\sqrt{d})=a-b\sqrt{d}$$ Since $\sqrt{d}\in \Reals$ it follows that $K$ is a totally real field.
2. Similarly, let $K=\Rats(\sqrt{d})$ with $d$ a square-free negative integer. Then $$\Sigma_K=\{ \operatorname{Id}_K, \sigma\}$$ where $\operatorname{Id}_K\colon K \hookrightarrow \Complex $ is the identity map ($\operatorname{Id}_K(k)=k$ for all $k\in K$ , whereas $$\sigma\colon K \hookrightarrow \Complex,\quad \sigma(a+b\sqrt{d})=a-b\sqrt{d}$$ Since $\sqrt{d}\in \Complex$ and it is not in $\Reals$ it follows that $K$ is a totally imaginary field.
3. Let $\zeta_n, n\geq 3$ be a primitive $n^{th}$ root of unity and let $L=\Rats(\zeta_n)$ a cyclotomic extension. Note that the only roots of unity that are real are $\pm 1$ If $\psi\colon L \hookrightarrow \Complex $ is an embedding, then $\psi(\zeta_n)$ must be a conjugate of $\zeta_n$ i.e. one of $$\{ \zeta_n^a \mid a \in (\Ints/n\Ints)^{\times}\}$$ but those are all imaginary. Thus $\psi(L)\nsubseteq \Reals$ Hence $L$ is a totally imaginary field.
4. In fact, $L$ as in $(3)$ is a CM-field. Indeed, the maximal real subfield of $L$ is $$F=\Rats(\zeta_n + \zeta_n^{-1})$$ Notice that the minimal polynomial of $\zeta_n$ over $F$ is $$X^2-(\zeta_n+\zeta_n^{-1})X+1$$ so we obtain $L$ from $F$ by adjoining the square root of the discriminant of this polynomial which is $$\zeta_n^2+\zeta_n^{-2}-2= 2\cos(\frac{4\pi}{n})-2 < 0$$ and any other conjugate is $$\zeta_n^{2a}+\zeta_n^{-2a}-2=2\cos(\frac{4a\pi}{n})-2 < 0, a\in (\Ints/n\Ints)^{\times}$$ Hence, $L$ is a CM-field.
5. Notice that any quadratic imaginary number field is obviously a CM-field.