Is there an impossible to solve equality in $\mathbb{C}$? exactly like how ${x^2=-1}$ is impossible in $\mathbb{R}$
is there any equation that is impossible in $\mathbb{C}$
and how to deal with ?

Sure, you can't solve equations like
$$xy-yx=1$$
in the complex plane. But if we extend the complex plane into quaternions then this equation does have a solution. The basis elements are usually written as $1,i,j,k$ so just like how $2+3i$ is a complex number, we can have $1+2i+3j+4k$ as a quaternion number. The reason we need four basis elements is because you can think of quaternions as two complex planes put together. There is no three-dimensional analogue in between because there is no meaningful way for us to extend the complex division into three dimensions. From two we go to four and then from four we have to go to eight (octonions) and then that's it.
Here is another answer of mine explaining this more in detail.
FYI, the above equations can be solved in quaternions because multiplication is not commutative anymore. So in general $xy\neq yx$ so we can find two numbers $x,y$ which can solve that equation above.

All non-constant polynomials with coefficients in $\mathbb{C}$ have a root in $\mathbb{C}$, so polynomial equations are solvable. There are, of course, equations that aren't polynomials that have no solutions, e.g.
$$e^{z} = 0$$
$$\frac{1}{z} = 0$$
and so on.