Does $a+ib=0$ imply $a=b=0$? If we have a complex number number $a + ib = 0$, then we can write as $a + bi = 0 + 0\cdot i$. Equating corresponding terms, we get $a = b = 0$.
What is this? Where am I going wrong ? 
 A: I don't quite understand your confusion, it is correct if a complex number $z = a+ib= 0$ (with $a,b\in\mathbb{R}$), then both its real part and its imaginary part are zero, i.e. $\text{Re}(z)=a=0$, and $\text{Im} (z) = b = 0$.
A: A complex number, Z, has the form $Z = x+iy$ where $x$ and $y$ are real numbers and $i$ is the imaginary unit postulated such that $i$$^2$ = $-1$.
Quantity $x$ is the real part of $Z$ and $y$ is the imaginary part of $Z$. Please note that $y$ is a Real quantity. 
Therefore, A Real Number is thus a complex number with $0$ Imaginary Part. 
A complex number with $0$ Real Part is a pure imaginary number. 
The number which is both Real and Pure imaginary at the same time is $0$. 
It is, therefore, trivial to write $0$ as $0$ + $i0$ which hardly serves any meaningful purpose except for the sake of brevity. With this background, you are going wrong NOWHERE.
A: "If we have a complex number number a+ib=0, then we can write as a+bi=0+0⋅i. Equating corresponding terms, we get a=b=0"
Yes, that is correct.  $0 = 0 + 0i.  Re(0) = 0.  Im(0) = 0$.  If $z = a + bi = 0 + 0i$ then $a = Re(0) = Re(0 + 0i) = 0; b = Im(0) = Im(0+0i) = 0;$ and $a = b = 0$.
I suppose part of the confusion is if $x \in \mathbb R \subset \mathbb C$ then $x = x + 0i$ and $Re(x) = x$ and $Im(x) = 0$ and if $x = a + bi$ then $a =x$ and $b = 0$.  But ... that shouldn't be confusing.  As a real number $x$ is "purely" real its imaginary component must be $0$.
.
"What is this?"
Um... A sentence????
"Where am I going wrong ? "
Nowhere.
A: This amounts to say that $1$ and $i$ are linearly independent, i.e. the only solution of
$$\lambda\cdot1+\mu\cdot i=0$$ is $$\lambda=\mu=0.$$

We can prove it by multiplying the equation by $i$ to form a system
$$\begin{cases}\ \ \ \lambda+\mu\, i=0\\-\mu+\lambda\,i=0.\end{cases}$$
It has a nonzero determinant ($2i$), which justifies the claim.
