No, it is not possible for two people to claim the same dependent and "split" the benefits in this way. When two individuals try to claim the same dependent, the IRS will only allow one person the claim by applying Tiebreaker rules. (Click on the link for more information).
In this case, if you both are biological parents of the child and lived together all year, the person with the higher claim is the parent with the higher income.
However, either of you can claim the child. You can each prepare your return with the child, and determine who gets the greater benefit. The other simply doesn't make any claim.
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