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I have in mind the result of

ComplexPlot3D[Log[MittagLefflerE[1/2, z]],{z, -50 - 50*I, 50 + 50*I}, 
PlotPoints -> 300,PlotRange -> All]

enter image description here

In particular, the plot should be seen over the whole quadrat.

Addition 1. Up to @J.M. is away sugestion,

ComplexPlot3D[Log[MittagLefflerE[1/2, z]], {z, -50 - 50*I, 50 + 50*I},
   PlotPoints -> 300, PlotRange -> All,   WorkingPrecision -> 25] // AbsoluteTiming

{874.714,enter image description here }

Addition 2. The command

ComplexPlot3D[MittagLefflerE[1/2, z], {z, -50 - 50*I, 50 + 50*I},PlotRange -> All]

performs an empty plot.

Addition 3. Following another suggestion by @J. M. will be back soon,

ComplexPlot3D[Log[MittagLefflerE[1/2, z]], {z, -50 - 50*I, 50 + 50*I},
 PlotPoints -> 300, PlotRange -> {0, 200}, WorkingPrecision -> 30]

enter image description here

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  • $\begingroup$ Some of it is due to large intermediate results not being computed accurately in machine precision; try setting WorkingPrecision -> 25, and there should be less holes in the surface. $\endgroup$ – J. M.'s discontentment Sep 15 '19 at 8:44
  • $\begingroup$ @J. M. is away: Thank you for your useful suggestion. $\endgroup$ – user64494 Sep 15 '19 at 9:08
  • $\begingroup$ You could try increasing the setting of both WorkingPrecision and PlotPoints further, but of course this will make your plots take longer to generate. $\endgroup$ – J. M.'s discontentment Sep 15 '19 at 9:23
  • $\begingroup$ @J.M. will back soon: All that is not so simple. In particular, the argument of the function under consideration is not constant as the above plots show. Also the result of ComplexPlot3D[Log[MittagLefflerE[1/2, z]], {z, -50 - 50*I, 0 + 0*I}, PlotPoints -> 300, PlotRange -> All] is not correct. $\endgroup$ – user64494 Sep 15 '19 at 9:29
  • $\begingroup$ "the result ... is not correct" - even after adjusting WorkingPrecision? $\endgroup$ – J. M.'s discontentment Sep 15 '19 at 9:32
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The branch cut structure here is very dense, which can be very hard for plotters to pick up.

Here are the cuts in a part of your domain:

Quiet @ Show[
  ComplexPlot[
    Log[MittagLefflerE[1/2, z]], {z, 0, 20 + 20 I}, 
    Exclusions -> None, ColorFunction -> {Automatic, None}, ImageSize -> Large
  ],
  ContourPlot[
    Im[E^(x + I y)^2 Erfc[-x - I y]] == 0, 
    {x, 0, 20}, {y, 0, 20}, 
    RegionFunction -> Function[{x, y, f}, Re[E^(x + I y)^2 Erfc[-x - I y]] <= 0], 
    PlotPoints -> 200, MaxRecursion -> 1, ContourStyle -> Black
  ]
]

enter image description here

We can turn these off with Exclusions -> None:

ComplexPlot3D[
  Log[MittagLefflerE[1/2, z]], 
  {z, -50 - 50*I, 50 + 50*I},
  Exclusions -> None, 
  NormalsFunction -> None,
  PlotRange -> All, 
  WorkingPrecision -> 25
]

enter image description here

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