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.

$$\Big(\frac{i+1}{\sqrt{2}}\Big)^2=\frac{(i+1)^2}{2}=\frac{1-1+2i}{2}=i$$
In general $\mathbb{C}$ is algebraically closed (fundamental theorem of algebra). So square root of every complex number is a complex number. You can find square root of $a+ib$ by solving $$(x+iy)^2=a+ib$$ (Eqaute the real and imaginary parts)
