Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$.
Quick web search gave no satisfactory results. Yet I believe this couldn't be considered trivial, could it?
I am contemplating quite a few ideas for example looking for possible kernels of homomorphisms. Maybe you could give some references? Ideally I'd love to figure out even multiple different proofs for this.
Fields only have the two obvious ideals. That is because in a field, every non-zero element is invertible, and invertible elements generate the entire ring as ideals.
I think you are getting confused by the terminolgy. Neither ${\mathbb R}$ nor ${\mathbb C}$ is a number field. A number field is a subfield of ${\mathbb C}$ which is a finite extension of ${\mathbb Q}$, such as ${\mathbb Q}(i)$ or ${\mathbb Q}(\sqrt[3]{2})$.
As others have pointed out, fields only have two ideals. But every number field $K$ has a ring of integers $D$ (which is the subring of the algebraic integers in the field), and it is customary within the algebraic number theory community to refer to ideals of $D$ as ideals of the number field. To further complicate matters, there are fractional ideals, which are finitely generated $D$-submodules $M$ of $K$ with the property that $cM \le K$ for some $c \in D$, and these are sometimes referred to simply as ideals.
It is known that a ring $R$ is a field iff it contains only trivial ideals, i.e., the only ideals in $R$ are $\{0\}$ and $R$ itself. See, e.g., this post.
