Okay, the title is a bit of a clickbait, I just want to know what properties of valuations extend to $\mathbb R$...

Denote an extension of the 2-adic valuation from $\mathbb Q$ to $\mathbb R$ by $\nu$. Suppose $\nu(x)=\nu(y)=0$.

Is it true that $\nu(x+y)\ne 0$?

What about $\nu(x^2+y^2)\le 1$?

I'm interested in knowing both whether these are true for every extension, as well as knowing whether there is some extension for which they are true (for every $x$ and $y$).