What is the scriptural basis for the idea that salvation can still be obtained after death? What is the basis in the Scripture for the doctrine that those who haven't received salvation during their physical life, especially those who heard of Christ and yet chose not to pray to Him, will still have a chance to get saved after their physical death?
The only passage I am aware of is 1 Peter 3:18-20a:


  For Christ died for sins once for all, the righteous for the unrighteous, to bring you to God. He was put to death in the body but made alive by the Spirit, through whom also he went and preached to the spirits in prison who disobeyed long ago when God waited patiently in the days of Noah while the ark was being built.


It should be noted that this is rather a "standalone" passage, in that there is nothing similar in Scripture, so it is unwise to base a doctrine on it. However, you could read into this:


Jesus is here preaching to dead people in hell (implied by "he went and preached to the spirits in prison who disobeyed") and
He would not preach unless there was hope of their salvation.


Note though that this is speculation and not a central belief of any mainstream Christian denomination (that I know of - but see @brillant's comments, below).