Once you collect the Chaos Crystal by buying it from the Serendipity timeline, bring the Chaos Crystal back to Hope in Academia AF4XX timeline. He’ll take the crystal and then tell you to check in with the clerk at the front desk.
Do so and you’re offered one of two weapons: The Odinblade for Noel, or the Odinbolt for Serah. Choose whichever you want, and then return to the Serendipity timeline. Check the shop and you can now buy the other chaos weapon for 750 gil.
Once you’ve obtained all 160 fragments, these two chaos weapons have an amazing 220 value in both STR and MAG. Unfortunately they offer no other perks.