$\mathbb{C}^\times$ mod roots of unity isomorphic to $\mathbb{C}^\times$ 
Let $H = \langle i \rangle =\{ i, -1, -i, 1 \}\le \mathbb{C}^\times$. 
  Then is $\mathbb{C}^\times/H$ isomorphic to $\mathbb{C}^\times$?

I don't think there exists an isomorphism 
$\varphi : \mathbb{C}^\times \to \mathbb{C}^\times/H$
because for any $z = re^{i\theta} \in \mathbb{C}^\times$, we can consider $z' = re^{i(\theta+\pi/2)}$ which gives $zH = z'H$. So this makes me think an isomorphism property can be broken somehow but I can't seem to make it work out. Or perhaps the quotient group is isomorphic to the multiplicative group of complex numbers.

Let $n$ be any positive integer, and $\mu_n$ the group of complex $n$-th roots of unity. The map $\Bbb C^\times\to\Bbb C^\times$ given by $z\mapsto z^n$
is a surjective group homomorphism with kernel $\mu_n$. By the First
Isomorphism Theorem, $\Bbb C^\times\cong\Bbb C^\times/\mu_n$.
This is the $n=4$ case.
