Let $K$ be an algebraic number field, $\mathcal{O}$ its ring of integers, $\mathfrak{m} \subset \mathcal{O}$ an integral ideal, and $J^{\mathfrak{m}}$ the group of integral ideals in $\mathcal{O}$ that share no factors with $\mathfrak{m}$.
On p. 470 of the English translation of $\textit{Algebraic Number Theory}$, Neukirch gives the following definition of a Größencharakter (What I think is more commonly referred to as a Hecke character):
$\textbf{(6.1) Definition.}$ A $\textbf{Größencharakter}$ mod $\mathfrak{m}$ is a character $\chi: J^{\mathfrak{m}} \to S^1$ for which there exists a pair of characters $$\chi_{\textrm{f}}: (\mathcal{O}/\mathfrak{m})^{\times} \to S^1, \qquad \chi_{\infty}:\mathbb{R}^{\times} \to S^1$$ such that $$ \chi((a)) = \chi_{\textrm{f}}(a)\chi_{\infty}(a)$$ for every algebraic integer $a \in \mathcal{O}$ relatively prime to $\mathfrak{m}$.
But in general, we do not have $\mathcal{O} \subset \mathbb{R}$ - e.g. when $K=\mathbb{Q}(i)$.
So if $a \in \mathcal{O}$, how are we to understand $\chi_{\infty}(a)$? Is it perhaps evaluated at its absolute value, or its image under the field norm?
Thank you for your attention.
Neukirch's $\mathbf R$ (not $\mathbb R$!) is the Minkowski space of $K$, i.e. $\mathbf R = K \otimes_{\mathbb Q} \mathbb R$. There is a natural embedding $K \hookrightarrow \mathbf R$. So if $\chi_\infty$ is a character of $\mathbf R^\times$, then you can evaluate it on $a \in K^\times$ via this embedding.
To give you an idea of what $\mathbf R$ looks like: it is isomorphic as a topological $\mathbb R$-algebra to $\mathbb R^{r_1} \times \mathbb C^{r_2}$, and $K$ is dense in it (when embedded in the natural way).
Examples: when $K = \mathbb Q$ we have $\mathbf R = \mathbb R$. When $K = \mathbb Q(i)$ we have $\mathbf R = \mathbb C$.
You are correct that Hecke character is a synonym of Grössencharacter. It's just that Hecke didn't name his characters after himself when he introduced them.