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)?

Proprietà globali degli spazi analitici reali, link.springer.com/article/10.1007%2FBF02416802 $\endgroup$ – Francesco Polizzi Apr 21 '17 at 12:25