spiral1=NDSolve[{x'[t]==y[t],y'[t]==-x[t],z'[t]==-z[t],x[0]==1,y[0]==1,z[0]==1},{x[t],y[t],z[t]},{t,0,100},MaxSteps->Infinity];

ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.spiral1],{t,0,20},PlotPoints->10000,BoxRatios->{1,1,1}]

α=-.5;

spiral2=NDSolve[{x'[t]==α x[t]+y[t],y'[t]==-x[t]+α y[t],z'[t]==z[t],x[0]==.01,y[0]==.01,z[0]==.01},{x[t],y[t],z[t]},{t,-100,100},MaxSteps->Infinity];

ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.spiral2],{t,-30,30},PlotPoints->10000,BoxRatios->{1,1,1},PlotRange->{{-2,2},{-2,2},{0,10}}]