Aims. We revisit the carbon abundance in the primitive iron-poor star J0815+4729 ([Fe/H] = -5.49). J0815+4729 is so enhanced in carbon and rich in oxygen that, in addition to the CH and C2 molecular lines, weaker neutral atomic transitions are measurable in its spectrum. This provides a unique opportunity to explore the line formation of C I and O I transitions in such an extremely iron-deficient star. Methods. Based on high-resolution spectroscopy and newly constructed atomic models for carbon and oxygen, we present the first investigation into deriving the carbon and oxygen abundances of this hyper-iron-poor star from neutral atomic transitions under nonlocal thermodynamic equilibrium (NLTE) conditions. Results. The carbon abundance we derived from C I lines is A(C)NLTE = 8.08 ± 0.07, which is consistent with that inferred from C2 lines (A(C)LTE = 7.96 ± 0.08), but higher by about 0.6 dex than the value from CH lines (A(C)LTE = 7.47 ± 0.17). Since the CH lines are the most commonly used carbon abundance indicator for extremely metal-poor stars, this result casts doubt on the quantitative interpretation of CH absorption in the most primitive stars. We confirm that J0815+4729 is one of the most carbon-enhanced stars known to date, with [Fe/H] <= -4.5. We also determined A(O)NLTE as 7.25 ± 0.11, with NLTE corrections of about -0.11 dex. The carbon-to-oxygen ratio of the star is [C/O] = 1.06 ± 0.11.