I have gout, can I take colchicine with indocin (indomethacin) together?

Yes. I know of no drug-drug interaction between these 2 medications. You can always have a pharmacist check their computers. But the two medications listed work differently and hence there is no reason why they can't.
Yes. Work in different ways to address the problem.
No. It is not recommended that you take the 2 drugs together.