Hilger real part
Let $h>0$ and let $z \in \mathbb{C}_h$, the Hilger complex plane. The Hilger real part of $z$ is defined by $$\mathrm{Re}_h(z)=\dfrac{|zh+1|-1}{h}.$$
Properties
Theorem: The following inequality holds for $z \in \mathbb{C}_h$: $$-\dfrac{1}{h} < \mathrm{Re}_h(z) < \infty.$$
Proof: █
Theorem
The following formula holds: $$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$ and $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$.
Proof
References
Theorem
The following formula holds: $$z = \mathrm{Re}_h(z) \oplus_h \mathring{\iota} \mathrm{Im}_h(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$, $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$, $\oplus_h$ denotes the circle plus operation, and $\mathring{\iota}$ denotes the Hilger pure imaginary.
