Interpreting magnitude of DFT results
One common practice is to take the log of the squared magnitude of the DFT frames, as in log(re*re+im*im), scale that, and use the scaled log as an integer index into an RGB color lookup table ("hotter" colors for larger magnitudes).  The scale is determined so that the range in your data fits within the color index table.

A log scale is used, since it's a closer match to human perception of loudness than a linear scale.

A square root is un-needed, as that's the same as adjusting the scale after the log by 0.5.