First off Gamma is 1/sqrt(1-v^2/c^2) but that's beside the point.

While by itself it looks okay, if you plug i into a Lorentz transformation everywhere there's a velocity you will also need to either have your time component complex or one of your spatial components as a complex number. This results in that component reversing sign meaning that the proper time or proper separation between events will no longer be equal which was the whole point of a Lorentz transformation in the first place.

Of note is that occasionally time is defined as complex for calculations in general relativity but this is done for both T and T' (I assumed that you were only defining it for one side of the equation), while this does result in what appears to be an imaginary velocity it's just a mathematical trick so the physicist doesn't have to bother keeping the signs straight in the calculations, since at the end it does not change the physics.

First off Gamma is $\frac{1}{\sqrt{1-v^2/c^2}}$ but that's beside the point.

While by itself it looks okay, if you plug i into a Lorentz transformation everywhere there's a velocity you will also need to either have your time component complex or one of your spatial components as a complex number. This results in that component reversing sign meaning that the proper time or proper separation between events will no longer be equal which was the whole point of a Lorentz transformation in the first place.

Of note is that occasionally time is defined as complex for calculations in general relativity but this is done for both T and T' (I assumed that you were only defining it for one side of the equation); while this does result in what appears to be an imaginary velocity it's just a mathematical trick so the physicist doesn't have to bother keeping the signs straight in the calculations, since at the end it does not change the physics.