What is the square root of complex number i? Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I wrong?
EDIT: my question is rather related to nomenclature of definition, while square root of -1 defined as i, why not j defined as square root of i and k square root of j and  if those numbers have deeper meanings and usage as in quaternions theory.

The answer is that 

the square root of $i$ isn't defined.

That is, when we talk about the square root of something we are talking about the function
$$
x \mapsto \sqrt{x}.
$$
Being a function means that for each input, you have exactly one output. For real numbers the square root of a non-negative number $x$ is defined as the unique non-negative number $y$ satisfying that $y^2 = x$.
The thing is that we don't have a universally agreed upon definition of the square root of a complex number.
Now, you can of course consider the equation $y^2 = i$. This has two solutions:
$$
y = e^{\frac{2\pi i}{4}} \quad \text{or} \quad y = e^{5\frac{2\pi i}{4}}.
$$
If you wanted to define a square root, you would be faced with the question of which root would you pick. 
So in general it isn't good to be talking about the square root as having two values  because we would like to think about the square root as a function. One might talk about a square root.
For more on this, talk a look at the answers to this question: How do I get the square root of a complex number?
