Motivations and Terminology
The term "henselian field" is ambiguous. What I mean when I say that $K$ is a henselian field is that there exists a henselian DVR $R$, such that $K=Frac(R)$. What I mean when I say that $L$ is a complete field is that there exists a complete DVR $S$ such that $L=Frac(S)$.
Note that every complete field is henselian. Examples of complete fields are $\mathbb{Q}((t))$ (the field of formal Laurent series with coefficients in $\mathbb{Q}$), $\mathbb{Q}_p$ (the $p$-adics), and so forth.
When I try to think of henselian fields that are not complete, the ones that immediately come to mind are algebraic over some complete field. For example $\mathbb{Q}((t))(\sqrt{2})$, $\mathbb{Q}_p^{un}$ (the maximal unramified extension of $\mathbb{Q}_p$), and so forth.
Question
Is it true that for every henselian field $K$ there exists a subfield $L\subset K$ such that $L$ is complete and such that $K/L$ is an algebraic extension?
EDIT: I've re-emphasized this in the comments, but I think it is important to put this in the body of the question: both the term "henselian field" and "complete field" are used in many different contexts to mean different things.
Note that under the definitions above $\mathbb{R}$ does not constitute as a complete field. (This is because $\mathbb{R}$ is not the fraction field of a complete DVR.)
Also note that I do not consider $\mathbb{Q}((t^{1/n}))_{n\in\mathbb{N}}$ to be a henselian field. (This is because my definition of a henselian field requires it to be the fraction field of a henselian DVR, not a general henselian ring.)