I have recently come across with the multiplicative group formed by the union of the real and imaginary axes minus the origin of the complex plane. Surprisingly (at least to me) I couldn't find any information about this group online. I think it may be isomorphic to $$\mathbb{Z}_4\times\mathbb{R}^{+},$$where $\mathbb{R}^{+}$ is the group of positive reals, but I am not sure. Does anyone have information about this group? Does it have a special name or has it been studied in any context?
Thanks in advance!
Your guess is correct, and you can realise it as follows: the $n$-roots of unity (i.e. solutions of $x^n = 1$) always form a cyclic group with multiplication: in fact, if $x^n = 1$ then also $(x^n)^2 = 1 \cdot 1 = 1$, and so on. Generators of this cyclic group are called primitive roots of unity.
Your case is the special case when $n=4$, in which the roots are $+1, -1, i, -i$. Note that both $i$ and $-i$ are primitive.
Now, to build your group, you are just attaching a copy of $\mathbb{R}^+$ to each root, i.e. you're taking the direct product:
$$\mathbb{Z}_4 \times \mathbb{R}^+$$
which this time happens to have a nice realization as the axis of the complex plane (but the same construction works for any other $n$).