How to solve equations involving multivalued functions in complex domain?

Question

Does this equation have a solution in the complex domain?

$$ \sqrt{x+3} = 3 + \sqrt{x} $$

Squaring both sides gives $\sqrt{x}=-1$, which suggests the solution to be $x=1$. But:

1. How can the complex domain solution be a purely real number?
2. Does this have a link to the multivalued nature of the function ($\sqrt{x}$) in the complex domain?
3. What is the correct way to deal with such equations involving multivalued functions in the complex domain?

Thanks in advance.