Is it possible to put two functions in densityplot like in the plot?

I like Jason B.'s answer. Here is a quick to type in alternative:

ImageAdd[Image[f1], Image[f2]]

Note that the axes ticks and values get blurry.

I assume in other cases this would be better:

ImageAdjust[ImageAdd[Image[f1], Image[f2]]]

If you consider a DensityPlot of two functions to be similar to a Plot3D of two functions viewed from the top, then

DensityPlot[
 Max[ψ[-500, -500], ψ[500, 500]], {x, -1000, 1000}, {y, -1000, 1000}, PlotPoints -> 100, 
 ColorFunction -> "SunsetColors", PlotRange -> All, 
 Exclusions -> None]

gives you the expected output.

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This also works for functions with overlapping features:

DensityPlot[Max[Sin[x], Cos[x]], {x, 0, 2 Pi}, {y, 0, 2 Pi}, 
 PlotPoints -> 100, ColorFunction -> "SunsetColors", PlotRange -> All, Exclusions -> None]

The corresponding Plot3D for comparison:

Plot3D[{Sin[x], Cos[x]}, {x, 0, 2 Pi}, {y, 0, 2 Pi}, 
 ColorFunction -> "SunsetColors", Boxed -> False, ViewPoint -> Top]

However, if your expect a "DensityPlot of two function" to give you the total combined density, then using Plus instead of Max is the appropriate approach.

In more involved visualization situations one can use Piecewise to combine the two functions in one DensityPlot or create two DensityPlots, restrict each to a certain area using the RegionFunction option or by specifying a region (documentation examples), and then combine both using Show.