Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify this subset with $X$) and such that $\overline X\setminus X$ is a proper analytic subset (with no conditions on its codimension). 


In particular, given two (different) compactifications of $X$, they always contain biholomorphic dense open subsets. 


My question is: given two compactifications of $X$, are they necessarily bimeromorphic?


More precisely, does the biholomorphism between the two dense open subsets always extend to a global bimeromorphic map?


I guess the answer is no, but after a moment of reflection I cannot find any counterexample.
Perhaps, it would suffice to look at compactifications of $\mathbb C^2$... 



Non-bimeromorphic compactifications 
As you guessed, the answer is **no**. 


The following counterexample in dimension $2$ can be found in Vo Van Tan's paper [On the compactification of strongly pseudoconvex surfaces III](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0195&DMDID=dmdlog32), *Mathematische Zeitschrift* **195** (1987), 259-267. 


The Stein surface $\mathbb{C}^\* \times \mathbb{C}^\*$ admits two algebraic compactifications $M\_1$ and $M\_2$ which are *not bimeromorphic*. In fact, $M\_1=\mathbb{P}^1 \times \mathbb{P}^1$ is a rational surface whereas $M\_2$ is a $\mathbb{P}^1$-bundle over an elliptic curve.


