It is well known that the hyperbolic cosine $\cosh$ induces a bijection from $[0,+\infty)$ to $[1,+\infty)$, with its inverse given by $$ \forall t\in[1,+\infty),\quad\operatorname{arcosh}(t)=\ln(t+\sqrt{t^2-1}). $$
I am interested in the extension in the complex plane of this. It would be natural to set $$ \forall z\in\mathbb C,\quad\operatorname{arcosh}(z)=\ln(z+\sqrt{z^2-1}). $$
Yet many authors (including Wikipedia) define it as $$ \forall z\in\mathbb C,\quad\operatorname{arcosh}(z)=\ln(z+\sqrt{z+1}\sqrt{z-1}). $$
Wikipedia mentions that the second definition allows us to speak of $\arccos(z)$ for imaginary $z$. I don't find it very convincing, because I consider the square root defined on $\mathbb R_-$ as well. So I was not able to understand what the first definition really fails to achieve, and what advantage the second one really brings. Any help or reference would be appreciated.