It is specified as this:
$$(a,b)(c,d)=(ac-bd,ad+bc).\,$$
I don't see anywhere where the decision to have - these components $ac-bd$ and + these components $ad+bc$ was made. Or why it is like this. Wondering if one could explain why it's like this, without using or referencing $i$ or $\sqrt{-1}$, just using the idea of complex numbers as an ordered pair of reals.
On
Your question is unreasonable, since it is because someone (Hamilton, in this case) was trying to formalize the complex numbers, as things of the form $a+bi$ with $a,b\in\mathbb R$ and $i^2=1$, that that definition was obtained.
On
The elements of $\mathbb{R}^2$ with the element-wise addition and the multiplication defined above satisfies the field axioms (you can check it).
On
In the geometric interpretation of the complex numbers, $(a,b)$ is a point in the plane, the components of which are abscissa and ordinate.
Complex numbers are seen, in polar form, as a vector length and vector angle. The complex multiplication by $(0,1)$ is defined to be a rotation by a right angle. More generally, multiplication by a complex can be seen as the application of a similarity transform, i.e. the combination of a scaling and a rotation.
Now if you look at the product of two complex numbers as the combination of two similarity transforms, considering the rotation only, you need to enforce
$$(\cos\alpha,\sin\alpha)(\cos\beta,\sin\beta)=(\cos(\alpha+\beta),\sin(\alpha+\beta)),$$ as the combination of two rotations is a single rotation by the sum of the angles.
And recall,
$$(\cos(\alpha+\beta),\sin(\alpha+\beta))=(\cos\alpha\cos\beta\color{red}-\sin\alpha\sin\beta,\sin\alpha\cos\beta\color{red}+\cos\alpha\sin\beta).$$
This comes from the fact that every complex number $z=a+ib$ corresponds to a matrix of the form $$A=\begin{pmatrix}a & -b \\ b & a\end{pmatrix}.$$ Now take another matrix of this form, say $$B=\begin{pmatrix}c & -d \\ d & c\end{pmatrix}.$$ The product $$AB=\begin{pmatrix}a & -b \\ b & a\end{pmatrix}\begin{pmatrix}c & -d \\ d & c\end{pmatrix}=\begin{pmatrix}ac-bd & -(ad+bc) \\ (ad+bc) & ac-bd\end{pmatrix}.$$ You can again read of real and imaginary part of the product from this. If you are interested in why complex numbers correspond to such matrices, I can write a few lines about this fact here or you spend some minutes searching for it on the internet.