  In this article we establish the following results: Let $(X, B)$ be a dlt
pair, where $X$ is a $\mathbb Q$-factorial K\"ahler $4$-fold -- (i) if $X$ is
compact and $K_X+B\sim_{\mathbb Q} D\geq 0$ for some effective $\mathbb
Q$-divisor, then $(X, B)$ has a log minimal model, (ii) if $(X/T, B)$ is a
semi-stable klt pair, $W\subset T$ a compact subset and $K_X+B$ is effective
over $W$ (resp. not effective over $W$), then we can run a $(K_X+B)$-MMP over
$T$ (in a neighborhood of $W$) which ends with a minimal model over $T$ (resp.
a Mori fiber space over $T$). We also give a proof of the existence of flips
for analytic varieties in all dimensions and the relative MMP for projective
morphisms between analytic varieties.
