TITLE: Existence of a skew field with surjective inner derivations
QUESTION [7 upvotes]: In my research, I've come twice now towards a skew field $K$ that satisfies the following:
$$\text{for all non-central element $a$, the map }\quad x\mapsto ax-xa\quad\text{ is onto.}$$
I am hoping such a field does not exist but am unable to prove it.
Does this condition ring a bell to some of you ?
Any example of such a field in both zero and non-zero characteristic ?
Edit: reading the paper suggested by @TomDeMedts'comment made me realise that this is precisely Question 3 of http://www.ams.org/journals/proc/1961-012-03/S0002-9939-1961-0124355-6/ asked by G. Meisters in 1961.

REPLY [3 votes]: In The range of derivations on a skew field and the equation ax-xb=c, Journal of the Indian Math. Soc. 37 (1973), 61--69, P.M. Cohn shows the following :
Theorem (Cohn 1973). Given a field $K$ with centre $k$, there is an extension field $L$ (still with $k$ as centre) such that the equation $$ax-xa=c$$ has a solution for each $a\in L$ that is transcendental over $k$ and each $c\in L$.
In particular, taking a skew field $K$ having an algebraically closed centre (hence infinite dimentionnal over $k$ as @René notes) provides an $L$ for which every non-zero inner derivation is onto.