Actually, the same concept of “completing each other” is expressed by Krishna in the Bhagavad Gita. Krishna says that “mamaivamso jiva loke”…that all beings are but an expansion of Me, they come from Me, and part and parcel of Me. Of course this “Me” represents Krishna. Our very existence rests upon Krishna as we all are minute sparks emanating from the huge unlimited fire Krishna. Inherently, we possess the same qualities as the Lord. Qualities such as wanting to be the enjoyer, controller, proprietor, possessing free will, wanting to be a friend, a lover, a son, a parent etc…all these propensities are initially possessed by God and because we are in His image (minute parts), we also possess similar qualities as Krishna.
While we can find partners to complete us in this temporary physical reality, we also need a partner to complete us in the spiritual reality. That partner is none other than Krishna. This partnership of completing each other in the spiritual reality is based upon unconditional love and service. Otherwise, it cannot exist. Unlike the material counterpart where love and service is tainted by selfishly motivated desires, in the spiritual reality, this sense of completing each other is purely and solely driven by the desire to serve and love the counterpart (I love Krishna and Krishna loves me) with no selfish motives and with no end in sight. This expression of love is called bhakti and the practice of bhakti is called bhakti yoga.
In one sense, when a devotee and Krishna love each other, that love is no different in concept than the love people of this world profess. In one sense, this is the ultimate destiny and desire of man – to love and be loved so we can complete our lives as one whole (yet individually maintaining our identities). However, this feeling of "whole" is complete only if this love is dovetailed on the spiritual reality with Krishna everything else of this physical reality will eventually fade away in time no matter how grandeur or sublime the love may be.
Dovetail your loving propensities towards Krishna and when Krishna reciprocates we will feel complete and fulfilled!