A new type of numbers? Humans started to think about solution to the equation x+2=0, they invented therein negative numbers. And now what about the equation $\sqrt{x}$+1=0, we invent new type of number?
For example we denote those numbers as $\pm$1 such that those numbers follow a particular set of rules just like negative numbers (-1x-1=1... and all other rules for instance)
(I know about imaginary numbers)
 A: When we extend the range of numbers, we do it in a way such that the 'old' numbers are embedded in the now larger range of numbers, and all 'old' calculations hold in the new context. This works fine for $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C$.
Now if we want to find a new type of numbers $\mathbb W\supseteq \mathbb C$, where we can solve $\sqrt{x}+1=0$, we should still be able to have the following argument:
Assume $x$ is a solution of $\sqrt{x}+1=0$, then $\sqrt{x}=-1$ by definition of the additive inverse. The only sensible definition of $\sqrt{x}$ would be a number $y$, that solves $y^2=x$, so by squaring both sides of $\sqrt{x}=-1$, we reach $x=1$. But then our 'old' definition of $\sqrt{x}$ should apply, since $1\in\mathbb C$, and we get $\sqrt{x}=1$ and therefore $1+1=0$ which is already wrong in $\mathbb C$, so it can't hold in $\mathbb W$ as well.
We can conclude that there is no larger number range $\mathbb W$ that is compatible with the numbers and definitions we use and contains a solution to $\sqrt{x}+1=0$.
On another note, the definition of $\sqrt{x}$ for $x\in\mathbb C$ is pretty arbitrary, since the equation $y^2=x$ has two solutions whenever $x\neq 0$, so which do we pick? If we pick $\sqrt{1}=-1$, your equation would be solvable in $\mathbb C$, but by convention we don't.
