OK:
(%i5) radcan(exp(((t-2*%i)*x^2)/(4*t-2*%i))), gcd : 'red;
(%o5) %e^(((t-2*%i)*x^2)/(4*t-2*%i))
Too messy:
(%i6) radcan(exp(((t-2*%i)*x^2)/(4*t-2*%i))), gcd : 'spmod;
(%o6) %e^(((32*t^6-144*%i*t^5-240*t^4+200*%i*t^3+90*t^2-21*%i*t-2)*x^2)/(128*t^6-384*%i*t^5-480*t^4+320*%i*t^3+120*t^2-24*%i*t-2))
See also
http://www.math.utexas.edu/pipermail/maxima/2011/025233.html