The Main Conjecture(s) of Iwasawa theory are an essential tool for studying the arithmetical properties of special values of \(L\)-functions attached to motives. In this paper, the authors treat the \(MC\) for an imaginary quadratic field \(K\) both in the character-wise and the equivariant (i.e., taking into account the Galois action of an abelian extension \(F/K)\) setting. We don't recall the statements of the \(MC's,\) except to say that they relate ``above \(K_\infty\)'' certain \(p\)-adic analytic objects (here, ``zeta elements'' built from Euler systems of elliptic units) to certain algebraic objects (here, the cohomology groups of certain perfect complexes \(\displaystyle{\mathop R_{=}}\;\Gamma ({\mathcal O}_K [1/p], \Lambda (\eta) (1))\) and \(\displaystyle{\mathop R_{=}}\;\Gamma ({\mathcal O}_K [1/p\;{\mathfrak f}], \;\Omega (1)).\) Notation : \(K_\infty\) = the maximal multiple \({\mathbb{Z}}_p\)-extension of \(K,\) \(\Gamma = \text{Gal} (K_\infty/K) \simeq {\mathbb{Z}}^2_p\), \(\Lambda = \Lambda (\Gamma) \simeq {\mathbb{Z}}_p [\![ S, T ]\!]\), \({\mathfrak{f}}\) = a non zero ideal of \({\mathcal O}_K\), \(\eta\) = a character \(\text{Gal}(K({\mathfrak f}) / K) \to {\mathbb{C}}^\ast\), \({\mathcal G}_{\mathfrak f} = \text{Gal}(K ({\mathfrak f} p^\infty) / K)\), \(\Delta\) = the torsion subgroup of \({\mathcal G}_{\mathfrak f}\), \(\Omega = \Lambda ({\mathcal G}_{\mathfrak f}) \simeq {\mathbb{Z}}_p [\Delta] \;[\![ S, T]\!]\). The character-wise (resp. equivariant) \(MC\) will be called \(\Lambda\)-\(MC\) (resp. \(\Omega\)-MC). The \(\Lambda\)-\(MC\) (in a more classical language) was proved by \textit{K. Rubin} [Invent. Math. 103, No. 1, 25--68 (1991; Zbl 0737.11030)], but modulo a semi-simplicity assumption, i.e., \(p\) does not divide the orders of the characters involved. In the first part of the present paper, the authors reprove the \(\Lambda\)-\(MC\) for all primes. To get rid of the problem of non semi-simplicity, they follow the scheme of proof developed by \textit{A. Huber} and \textit{G. Kings} [Duke Math. J. 119, No. 3, 393--464 (2003; Zbl 1044.11095)]: instead of decomposing the classical Iwasawa modules under character-wise projectors (which may not be integral), they use Galois cohomology with coefficients in the Galois representations defined by the characters. Following \textit{K. Kato}'s treatment of the Euler system of elliptic units [Cohomologies \(p\)-adiques et applications arithmétiques. III. Paris: Société Mathématique de France. Astérisque 295, 117--290 (2004; Zbl 1142.11336)] and ``going up in the direction of \(K_\infty\)'', the usual machinery yields ``half'' of the \(MC,\) namely certain divisibility relations. To get the reverse divisibility, the authors use a precise relation between elliptic units and zeta elements to reduce the problem to the Tamagawa Number Conjecture at \(s = 0\) (a generalization of the analytic class number formula), proved for abelian fields e.g. in Huber-Kings [op. cit.]. In the second part of the paper, the authors follow the scheme of proof of \textit{M. Witte} over \({\mathbb Q}\) [Acta Arith. 122, No. 3, 275--296 (2006; Zbl 1098.11055)] to derive the \(\Omega\)-\(MC\) from the \(\Lambda\)-\(MC\) by assuming the (now familiar) ``\(\mu = 0\)'' hypothesis, which implies the vanishing of the localized \(H^2\) at the so called singular primes. Note that a result of \textit{R. Gillard} [J. Reine Angew. Math. 358, 76--91 (1985; Zbl 0551.12011)] implies ``\(\mu = 0\)'' for all primes \(p\nmid 6\) which split in \(K\).