Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part and complex part? For example, polar form describes the relation on a imaginary and real plane. What concept is rectangular form based on?
[Math] Rectangular form of a complex number
complex numberscomplex-analysis
Related Solutions
We have $a + bi$ is algebraic iff $a$ and $b$ are algebraic.
Therefore, if $a + bi$ is transcendental then at least one of $a$ or $b$ is transcendental.
So, all complex transcendental numbers are "based" on real transcendental numbers.
One way to think of the complex plane $\mathbb{C}$ is to think of it as the real coordinate (or $xy$-) plane $\mathbb{R}^2$. In $\mathbb{R}^2$, we have two real-numbered axes (in particular, the $x$ and $y$ axes). An element of $\mathbb{R}^2$ is an ordered pair $(x, y)$, where $x$ and $y$ range over all the real numbers $\mathbb{R}$.
What I am about to say is not exactly rigorous, but I purposefully omit rigor in favor of your understanding.
Now, what happens if we decide to multiply $y$ by $i$, for all $y$ in $\mathbb{R}$? We obtain $i \mathbb{R}$, where each element is of the form of $i y$, where $i$ is of course defined as usual and $y$ is a real number. Consequently our vertical $y$-axis now becomes imaginary, and we obtain the complex plane $\mathbb{C}$. Note that just like the real plane $\mathbb{R}^2$, $\mathbb{C}$ is 2-dimensional. To recap, $\mathbb{C}$ has two axes, one real axis and one imaginary axis.
What does an element of $\mathbb{C}$ look like? An element $z$ of $\mathbb{C}$ has the form $a + bi$, where $a$ and $b$ are real numbers; we call $z$ a complex number. If we were to graph a complex number $z = a + bi$, we would graph it just like we would an ordered pair $(a, b)$ in $\mathbb{R}^2$. E.g., the number $z = 1 + i$ would correspond to the point $(1, 1)$; the number $z = 1$ would correspond to the point $(1, 0)$. This second example demonstrates that a real number is in fact a complex number as well.
I am not sure if you have seen linear algebra or not, but your statement in the second sentence of your first paragraph can be explained with such ideas. In the real plane $\mathbb{R}^2$, we have two directions; namely, the $x$ and $y$ directions. It turns out that we only need the number $x = 1$ to generate the entire $x$-axis. Similarly we may do the same with the $y$-axis. Together, we obtain the whole plane $\mathbb{R}^2$. $\mathbb{C}$ carries the same idea, except your vertical axis is generated by $i = 1\cdot i$ instead of just $1$.
To further address your question, recall $i$ is defined as $i = \sqrt{-1}$. Thus, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$. What have we accomplished here? We have obtained directions for the vertical and horizontal axes. From here we may generate all of $\mathbb{C}$.
I am not entirely sure how to answer your next few questions, but recall from above that $\mathbb{C}$ is 2-dimensional. Thus, $\mathbb{C}^2 = \mathbb{C} \times \mathbb{C}$ would be 4-dimensional. Generalizing this notion, $\mathbb{C}^n$ is $2n$-dimensional. Hence I do not think there is really an analogy to the $z$ axis if we are referring to the third axis in $\mathbb{R}^3$. The best analogy I can come up with are several-dimensional complex number systems such as $\mathbb{C}^2$.
Best Answer
You have $\mathbb{R}$ which is a field, i.e. you can add and multiply, and every nonzero elements has a multiplicative inverse.
Now you want to somehow have square roots for all real numbers. So you include (this can be done properly, but it's not the point here) another "number", that we call $i$ and has the property that $i^2=-1$.
Now you still want to have a field; in particular you want to add and multiply, and so you need to make sense of expressions of the form $a+bi$. But it turns out that $$ \mathbb{C}=\{a+bi:\ a,b\in\mathbb{R}\} $$ is already a field, i.e. the smallest field that contains both $\mathbb{R}$ and $i$.