Does a Kähler manifold always admit a complete Kähler metric? Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case?

Does a Kähler manifold always admit a complete Kähler metric?

Of course, every metric on a compact manifold is complete, so the question is only of interest in the non-compact case.
One might hope that the proof in the aforementioned note will still be of use, but as is shown in this question, a metric conformal to a Kähler metric cannot be Kähler (with respect to the same complex structure) except in complex dimension one.
The only condition that I am aware of that ensures the existence of complete Kähler metrics is that the manifold is weakly pseuodoconvex (i.e. it admits a plurisubharmonic exhaustion function), see Demailly's Complex Algebraic and Analytic Geometry, Chapter VIII, Theorem 5.2.
 A: I give a related answer for the following non-compact case which we can get complete Kähler metric
Take $\overline M$ be a compact Kähler manifold and $Y\subset \overline M$ be the simple normal crossing divisor and take $M=\bar M\setminus Y$ now we can define complete Kähler metric $\omega_P$ on non-compact manifold $M$ as follows
Since $Y$ is simple normal crossing divisor , so it can be defined by the equation $z_1^{\alpha}\cdots z_{n_\alpha}^\alpha=0$
Take a cover for $\overline M=U_1\cup\cdots\cup U_p\cup \cdots \cup U_q$ such that $\overline{U_{p+1}}\cup\cdots \cup \overline{U_{q}}=\phi$
Let $\{η_i\}_{1≤i≤q}$ be the partition of unity
subordinate to the cover $\{U_i\}_{1≤i≤q}$. Let $\omega$ be a Kähler metric on $M$ and let $C$ be a positive
constant. Then for $C$ enough large, the following Kähler form is complete Kähler metric 
$$\omega_P=C\omega+\sum_{i=1}^p\sqrt{-1}\partial\bar\partial\left(\eta_i\log\log\frac{1}{z_1^{i}\cdots z_{n_i}^i}\right)$$
See the paper https://projecteuclid.org/download/pdf_1/euclid.jdg/1214448444
Moreover Let $X$ be a singular subvariety of the compact Kähler manifold $M$ and let $\omega$ be a Kahler $(1,1)$ form on $M$ then the Saper-form
$$\omega_{Saper}=\omega-\frac{\sqrt{-1}}{2\pi}\partial\bar\partial \log(\log F)^2$$
is a complete Kähler metric on $M-X_{sing}$
A: Grauert proved that a relatively compact domain with real analytic boundary in C^n has a complete Kahler metric iff the boundary is pseudoconvex .For any
relatively compact domain in C^n which is the interior of its closure ,with a complete Kahler metric, Diederich and Pflug showed that it is locally Stein.
 See the paper of Demailly in Annales ENS vol 15 1982 page 487 .
