Bowing one’s head and closing one’s eyes for prayer is a personal preference. There are some rituals in the Catholic Church that do call for bowing one’s head.
In ordinations and religious profession of vows the candidate is to bow his head to the bishop or to the Major Superior while hands are imposed. During the liturgy of the Christmas season we bow during the part of the Creed that proclaims that Christ was born of the Virgin Mary.
In some communities, when the Glory be . . . is said at the end of the psalms during the Liturgy of the Hours we bow.
Bowing among Catholics is usually part of ritual. It has a special meaning in each ritual. However, in informal prayer or private prayer settings, it is up to the individual. Some people need it to concentrate. I know that I do. I’m a very visual person. I tend to take in everything I see. I need to close my eyes.