Proving $\langle \Bbb R,+ \rangle \cong \langle \Bbb C,+ \rangle$ without choice

Question

In this question, we have proved that $\langle \Bbb R,+ \rangle \cong \langle \Bbb C,+ \rangle$:

1. Pick your favourite Hamel basis $H=\{U_\alpha \mid \alpha \in I\}$ where $I$ is an indexing set.
2. Then, $H \cup iH$ is a basis of $\Bbb C$ as a vector space over $\Bbb Q$.
3. Pick your favourite bijection $\varphi:H \mapsto H \cup iH$.
4. This gives an isomorphism between the two groups concerned.

Choice is used in the 1^st step and, I believe, in the 3^rd step.

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Let $P$ be the proposition $\langle \Bbb R,+ \rangle \cong \langle \Bbb C,+ \rangle$ under $ZF$. Which of the following statements are true?

1. $\vdash P$
2. $\vdash \neg P$
3. $\operatorname{Con}(ZF \cup \{P\})$
4. $\operatorname{Con}(ZF \cup \{\neg P\})$