We measure the Born cross sections of the process e(+)e(-) -> K+ K- K+ K- at center-of-mass (c.m.) energies, root s, between 2.100 and 3.080 GeV. The data were collected using the BESIII detector at the BEPCII collider. An enhancement at root s = 2.232 GeV is observed, very close to the e(+)e(-) -> Lambda(Lambda) over bar production threshold. A similar enhancement at the same c.m. energy is observed in the e(+)e(-) -> phi K+ K- cross section. The energy dependence of the K+ K- K+ K- and phi K+ K- cross sections differs significantly from that of e(+)e(-) -> phi pi(+)pi(-).