{{x + Cosh[2 t], -I, (x + Sinh[2 t])/Sqrt[2],


 0, -((-3 x + Sinh[2 t])/Sqrt[2]), 0}, {I, x + Cosh[2 t], 0, (



x + Sinh[2 t])/Sqrt[2], 0, (-3 x + Sinh[2 t])/Sqrt[2]}, {(



 x + Sinh[2 t])/Sqrt[2], 0, 1/2 (1 + x + Cosh[2 t]), -I,




 1/2 (1 + 3 x - Cosh[2 t]), 0}, {0, (x + Sinh[2 t])/Sqrt[2], I,




 1/2 (1 + x + Cosh[2 t]), 0,




1/2 (-1 - 3 x + Cosh[2 t])}, {-((-3 x + Sinh[2 t])/Sqrt[2]), 0,




 1/2 (1 + 3 x - Cosh[2 t]), 0,



 1/2 (1 + 9 x + Cosh[2 t]), -I}, {0, (-3 x + Sinh[2 t])/Sqrt[2], 0,



1/2 (-1 - 3 x + Cosh[2 t]), I, 1/2 (1 + 9 x + Cosh[2 t])}}

evs = Eigenvalues[m]

Plot3D[Evaluate@evs, {x, -2, 2}, {t, 0, 2},
  PlotRange -> {-10, 20},  AxesLabel -> {"x", "t"},
  PlotStyle -> Opacity[0.3], Mesh -> None
]