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We study a non-Hermitian (NH) $sl(2)$ affine Toda model coupled to fermionsthrough soliton theory techniques and the realizations of the pseudo-chiral andpseudo-Hermitian symmetries. The interplay of non-Hermiticity, integrability,nonlinearity, and topology significantly influence the formation and behaviorof a continuum of bound state modes (CBM) and extended waves in the localizedcontinuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalarfield topological charges and winding numbers in the spectral topology areuncovered.  The biorthogonal Majorana zero modes, dual to the NH Toda solitons withtopological charges $\frac{2}{\pi} \arg{(z=\pm i)}=\pm 1$, appear at thecomplex-energy point gap and are pinned at zero energy. The zero eigenvalue $\l(z = \pm i)=0$, besides being a zero mode, plays the role of exceptional points(EPs), and each EP separates a real eigenvalue ${\cal A}$-symmetric and ${\calA}$-symmetry broken regimes for an antilinear symmetry ${\cal A}\in \{ {\calP}{\cal T}, \g_5{\cal P}{\cal T}\}$. Our findings improve the understanding ofexotic quantum states, but also paves the way for future research in harnessingnon-Hermitian phenomena for topological quantum computation, as well as theexploration of integrability and NH solitons in the theory of topologicalphases of matter.

Biorthogonal Majorana zero modes, ELC waves and soliton-fermion duality in non-Hermitian $sl(2)$ affine Toda coupled to fermions