Bijective map from upper half plane and $x$-axis to the whole of $\mathbb{R} ^{2} $. Is it possible to find a bijective map from $ \{ (x,y) \in \mathbb{R} ^{2} : y \geq 0 \} $ to the whole of $\mathbb{R} ^{2} $ (or consider it $ \mathbb{C} $ if needed)?

Sure, you can just pick your favourite bijection from $\mathbb R$ to $\mathbb R^{\ge 0}$, to apply to the $y$ bit of your coordinates. If you'll accept the Schröder-Bernstein theorem, there exists such a bijection since $x \mapsto e^x : \mathbb R \to \mathbb R^{\ge 0}$ is an injection, and the inclusion map the other way is an injection, so there exists $f: \mathbb R^{\ge 0} \to \mathbb R$ bijective, and then $(x, y) \mapsto (x, f(y))$ is such a bijection.
We could explicitly define a bijection $f$ as something like
\begin{align*}
f: \mathbb R \times \mathbb R^{\ge 0} &\to \mathbb R^2 \\
(x, y) &\mapsto
\begin{cases}
 (x, \ln(y + 1)) & y \in \{0, 1, 2, \dotsc\} \\
 (x, \ln y) & \text{otherwise}
\end{cases}
\end{align*}
but it's just a bit messy.
