(* runtime: 37 seconds *)
n=345;Repair[z_]:=If[MachineNumberQ[z],z,$MaxMachineNumber];Hue2[h_,b_]:=Hue[h,Min[1,2 (1-b)],Min[1,2 b]];
SuperExp[z_]:=Repair[If[Re[z]≤-1,Log[SuperExp[z+1]],If[Re[z]>0,Exp[SuperExp[z-1]],t=1;sum=Log[z+2]+Log[E/2];Scan[(t*=z/2;sum+=# t)&,{1.18353470251664338875,1.58593285160678321155,1.36629265207672068172,1.36264601823980036066,1.21734246689515424045,1.10981816083559525765,0.9667469297476984913,0.84089872598668435888,0.71353210966804747617,0.60168548504001373445,0.49928574281440518678,0.41140086629121763728,0.33506195665178500898,0.27104779243942234146,0.21728554054610033086,0.17311050207880035456,0.13690016038526570119,0.10765949732729711286,0.08413804539743192923,0.06542450487497340761,0.05060001212013485322,0.03895655493977817629,0.02985084640296329153,0.02277908979501017117,0.01730960309240666892,0.01310389615589767874,0.00988251130733762764,0.00742735935367278347,0.00556296426263720549,0.00415334478103463346,0.00309116153137843543,0.00229387529664008653,0.00169729976398295653,0.00125245885041635465,0.00092172809095368547,0.00067661152429638357,0.00049544127485341987,0.00036192128589181518,0.00026376927786672476,0.0001918084004526757,0.00013917553105723647,0.00010077412023867018,0.00007281884753121133,0.00005251474516228446,0.00003779882770351268,0.00002715594536867241,0.00001947408515177282,0.00001394059355016322,9.96213949015693*^-6,7.1071387229271*^-6,5.06199803708578*^-6,3.59960968975399*^-6,2.55569149787694*^-6,1.81175810338313*^-6,1.2824583153843*^-6,9.0647322737496*^-7,6.3980422418981*^-7,4.5095738191441*^-7,3.1741772125007*^-7,2.2312521183625*^-7,1.5663840476155*^-7,1.098230101323*^-7,7.690305934973*^-8,5.378502675604*^-8,3.757126131521*^-8,2.621429405247*^-8,1.826909956818*^-8,1.271754463425*^-8,8.84310192977*^-9,6.14230041407*^-9,4.26177146865*^-9,2.95386817285*^-9,2.04522503591*^-9,1.41464900426*^-9,9.7750884878*^-10,6.7478454029*^-10,4.6535930671*^-10,3.2062550784*^-10,2.2069891976*^-10,1.5177557961*^-10,1.0428189463*^-10,7.158597119*^-11,4.90980671*^-11,3.364531769*^-11,2.303635851*^-11,1.575933679*^-11,1.077213757*^-11,7.35717912*^-12,5.02077719*^-12,3.42362421*^-12,2.33271256*^-12,1.58818623*^-12,1.08046566*^-12,7.3450488*^-13,4.9894945*^-13,3.3868911*^-13,2.2973789*^-13,1.5572383*^-13,1.0548054*^-13,7.139840*^-14}];sum]]];
Show[Graphics[RasterArray[Table[z=SuperExp[x+I y];Hue2[Mod[Arg[z]/(2 Pi),1],Max[0,0.4+Log[Abs[z]]/20]],{y,-2.5,2.5,10.0/n},{x,-5.0,5.0,10.0/n}]]]]