Square root of Matrix $A=\begin{bmatrix} 1 &2 \\ 3&4 \end{bmatrix}$ Find Square root of Matrix $A=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$ without using concept of Eigen Values and Eigen Vectors
I assumed its square root as $$B=\begin{bmatrix} a &b \\   c&d  \end{bmatrix}$$
hence
$$B^2=A$$ $\implies$
$$\begin{bmatrix} a^2+bc &b(a+d) \\   c(a+d)&d^2+bc  \end{bmatrix}=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$$
So
$$a^2+bc=1 \tag{1}$$
$$b(a+d)=2 \tag{2}$$
$$c(a+d)=3 \tag{3}$$
$$d^2+bc=4 \tag{4}$$  From $(2)$ and $(3)$ we get
$$\frac{b}{c}=\frac{2}{3}$$
Let $b=2k $ and $c=3k$ Then
$$a^2=1-6k^2$$ and $$d^2=4-6k^2$$  So
$$B=\begin{bmatrix} \sqrt{1-6k^2} &2k\\   3k&\sqrt{4-6k^2}  \end{bmatrix}$$ 
So
$$B^2=\begin{bmatrix} 1 &2k \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right) \\   3k \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right)&4  \end{bmatrix}=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$$
Now we have to solve for $k$ using equation
$$ \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right)=\frac{1}{k} \tag{5}$$ Also
$$\left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right) \times \left(\sqrt{4-6k^2}-\sqrt{1-6k^2}\right)=3$$ So from $(5)$
$$\left(\sqrt{4-6k^2}-\sqrt{1-6k^2}\right)=3k \tag{6}$$
Subtracting $(6)$ from $(5)$ we get
$$2\sqrt{1-6k^2}=\frac{1}{k}-3k$$  Squaring Both sides we get
$$33k^4-10k^2+1=0$$ we get
$$k^2=\frac{5 \pm i \sqrt{8}}{33}$$ From this we get $k$. is there any other simpler way to find ?

Square roots suck, so I'll do this with polynomials instead. Any matrix $B$ with $B^2=A$ must commute with $A$. Since there exist vectors $e$ such that $(e,Ae)$ form a basis of $\Bbb R^2$ (in fact the first standard basis vector will do, as will almost any other vector), any matrix commuting with $A$ must be a polynomial in$~A$, as shown for instance elementarily in this answer.
Since $A^2=2I_2+5A$ (check it) any polynomial in$~A$ can be rewritten to a polynomial of degree${}<2$ in$~A$. So let $B=sI_2+tA$ then $B^2=s^2I_2+2stA+t^2A^2=(s^2+2t^2)I_2+(2st+5t^2)A$. You need that to be equal to $A$ and since $I_2$ and $A$ are linearly independent, it will hold if and only if $s^2+2t^2=0$ and $2st+5t^2=1$. Clearly there are no solutions over $\Bbb R$, so if the question was to find a real square root of$~A$, the answer is there are none.
Working over the complex numbers, we can however continue. We must have $t\neq0$, and then the first equation says $(s/t)^2=-2$. First take $s/t=\sqrt2\,\mathbf i$, then the second equation becomes $\sqrt8\,\mathbf it^2+5t^2=1$ or $t^2=\frac{5-\sqrt8\,\mathbf i}{33}$. Taking complex square roots gives you two solutions. Taking instead $s/t=-\sqrt2\,\mathbf i$ one gets $t^2=\frac{5+\sqrt8\,\mathbf i}{33}$ with again two solutions. These appear to be the ones you found in the question. I don't think it will clarify anything to detail these solutions any further (it just becomes a mess, though it will work if you persevere).
