basically normal and weight are relative terms.

I am consiering the example of a man standing on a weighing machine accelerating upwards .

Now the forces on the man w.r.t the elevator frame are the weight downwards pseudo force(ma) downwards and a normal force upwards(force exerted by the machine). from this we get N=mg+ma. Now this reading of N will be shown by machine and would be the apparent weight of man.

P.S Mass remains same but weight can differ