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Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?

By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to T_{f(x)}N$ becomes an isomorphism for every $x\in M$. It seems that using the existence of analytic structure on any smooth manifold should imply this (Can every manifold be given an analytic structure?) but since I can't find a reference I am not sure whether this is true or not. So my questions are

Q1. Does $N$ always exist?

Q2. If it does, is it unique in some sense?

Q3. Where can I learn more about this (if this is already known)?

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    $\begingroup$ By a result of Tognoli, it follows that any real analytic space admits a Stein complexification. See A. Tognoli, Proprietà globali degli spazi analitici reali, link.springer.com/article/10.1007%2FBF02416802 $\endgroup$
    – Francesco Polizzi
    Commented Apr 21, 2017 at 12:25
  • $\begingroup$ @FrancescoPolizzi Thanks for your reply. Is there an english translation of the article mentioned. Also I am interested more in the case where $N$ is compact (assuming M is compact), but maybe this doesn't exist $\endgroup$
    – Omar
    Commented Apr 21, 2017 at 12:39
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    $\begingroup$ unfortunately, I do not think any english translation is available. $\endgroup$
    – Francesco Polizzi
    Commented Apr 21, 2017 at 12:46
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    $\begingroup$ A reference in English of the result that Francesco mentioned above is this: link.springer.com/chapter/10.1007%2F978-3-322-84243-5_3 $\endgroup$
    – HYL
    Commented Apr 21, 2017 at 12:56
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    $\begingroup$ another paper on this topic is download.springer.com/static/pdf/647/…? $\endgroup$
    – Xiaoyang Chen
    Commented Apr 27, 2017 at 5:44

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