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This seems to work fine:

sage: quat_alg = QuaternionAlgebra(ext_alg,-1,-1)
sage: ext_alg.<dx0,dx1,dx2,dx3> = ExteriorAlgebra(quat_alg)
sage: dx = x0+x1*i+x2*j+x3*k
sage: dx_dag = x0-x1*i-x2*j-x3*k
# now some computations
sage: dx*dx_dag
-2*i*x0^x1 - 2*j*x0^x2 - 2*k*x0^x3 - 2*k*x1^x2 + 2*j*x1^x3 - 2*i*x2^x3
sage: dx*dx_dag*dx
-6*k*x0^x1^x2 + 6*j*x0^x1^x3 - 6*i*x0^x2^x3 + 6*x1^x2^x3
sage: dx*dx_dag*dx*dx_dag
-24*x0^x1^x2^x3

This seems to work fine:

sage: quat_alg = QuaternionAlgebra(ext_alg,-1,-1)
QuaternionAlgebra(QQ,-1,-1)
sage: ext_alg.<dx0,dx1,dx2,dx3> = ExteriorAlgebra(quat_alg)
sage: dx = x0+x1*i+x2*j+x3*k
sage: dx_dag = x0-x1*i-x2*j-x3*k
# now some computations
sage: dx*dx_dag
-2*i*x0^x1 - 2*j*x0^x2 - 2*k*x0^x3 - 2*k*x1^x2 + 2*j*x1^x3 - 2*i*x2^x3
sage: dx*dx_dag*dx
-6*k*x0^x1^x2 + 6*j*x0^x1^x3 - 6*i*x0^x2^x3 + 6*x1^x2^x3
sage: dx*dx_dag*dx*dx_dag
-24*x0^x1^x2^x3

This seems to work fine:

sage: quat_alg = QuaternionAlgebra(QQ,-1,-1)
sage: ext_alg.<dx0,dx1,dx2,dx3> ext_alg.<x0,x1,x2,x3> = ExteriorAlgebra(quat_alg)
sage: dx x = x0+x1*i+x2*j+x3*k
sage: dx_dag x_dag = x0-x1*i-x2*j-x3*k
# now some computations
sage: dx*dx_dag
x*x_dag
-2*i*x0^x1 - 2*j*x0^x2 - 2*k*x0^x3 - 2*k*x1^x2 + 2*j*x1^x3 - 2*i*x2^x3
sage: dx*dx_dag*dx
x*x_dag*x
-6*k*x0^x1^x2 + 6*j*x0^x1^x3 - 6*i*x0^x2^x3 + 6*x1^x2^x3
sage: dx*dx_dag*dx*dx_dag
x*x_dag*x*x_dag
-24*x0^x1^x2^x3