Will supercritical nitrogen stay supercritical if we raise temperature and pressure high above critical values?
If we maintain nitrogen at 220 bar and 400 °C, what will be the state of $\ce{N2}$? Will it be still supercritical?


Yes, it will still be supercritical.  The only reasonable definition of "supercritical" is something along the lines of "a fluid phase without surface tension wherein the fluid pressure and temperature exceed the critical point".

Thus, compressed air is a supercritical fluid.  The same is true of compressed nitrogen.

You may be wondering how "supercritical" air is different from regular, non-supercritical, gaseous air.  The answer is, it isn't meaningfully different.

If you research "supercritical fluids", it will be very hard to discover this fact.  That is because the "exotic" properties of "supercritical fluids", such as having densities "in between that of a gas and a liquid", are really only properties of supercritical fluids that are near (i.e. slightly above) the critical point.

It isn't really much of an exaggeration to say that supercritical fluids are really gases.  They have no surface tension.  They expand to fill their container.  Just like gases.  Near the critical point, supercritical gases can have densities that are very unusual for a gas. At these high densities, the gas is very non-ideal, and intermolecular interactions in the gas lead to all sorts of interesting behavior.  Those high-density intermolecular interactions give rise to the interesting properties that people refer to when they say "supercritical fluid".  The strength and nature of those interesting intermolecular interactions depends strongly on density, which is why people refer to "supercritical fluids" having "tunable" properties.  Small changes in density (by changing temp. or pressure) can give rise to large changes in those interactions.

At conditions way past the critical point, those intermolecular interactions don't matter very much any more, and the "supercritical fluid" again starts to behave more and more like an ideal gas.


  I know that phase diagram says after critical temperature and pressure we have a supercritical phase, but with enough p and T, more exotic states of matter occur.


Yes, other answers and comments have done a good job of discussing this point.  If you increase the pressure enough, eventually you'll have a black hole!  Of course, long before that happens, you can get other less exotic phases of matter such as various solid phases.  In general, predicting the high-pressure phase behavior of even small molecules like nitrogen is very difficult, and as such, experimental data is required.  Unfortunately, measuring the high-pressure phase behavior of materials is also very difficult, at least past a few hundred gigapascals.  So I'm not sure where the high-pressure "exotic" phases of e.g. solid nitrogen start to form.  But it's not at 220 bar and 400 &deg; C, under those conditions nitrogen will just be a gas, albeit a "supercritical" gas.