I hope this question ist still interesting for some people, as it is for me. I will first try to give a motivation for the "ghost map", i.e. the Witt polynomials, which, I think, is absent from Harder's article, and then give some historical remarks.
Motivation (partly inspired by the impressive article on Witt vectors in the German wikipedia): For this forget for a moment that we know the Teichmüller representatives. Imagine we have a strict $p$-ring $A$ with (perfect) residue ring $k$, that is, for any set-theoretic section $\sigma: k \rightarrow A$ of $A \twoheadrightarrow k$, every element of A can be written as
$$\sum_{i=0}^\infty \sigma(a_i) p^i$$
with unique $a_i$ in $k$. That's a set-theoretic bijection
$$k^\mathbb{N} \leftrightarrow A .$$
How to describe the ring structure on the "coordinates" on the left? It suffices to describe it mod $p^{n+1}$ for every n. The most naive "coordinate map" would be
$$k^{n+1} \rightarrow A/p^{n+1} A$$
$$(a_0, ..., a_n) \mapsto \sigma(a_0) + \sigma(a_1) p + ... + \sigma(a_n)p^n .$$
To this, there would correspond the naiveWitt polynomial
$$NW(X_0, ..., X_{n}) = X_0 + pX_1 + ... + p^n X_n .$$
"Naive" because the above map induced by it depends on $\sigma$. (We could even do worse and choose different $\sigma$'s for each coordinate, still maintaining a bijection). Now do remember -- not the Teichmüller representatives, but the crucial fact from their construction: $a \equiv b$ mod $p \Rightarrow a^{p^i} \equiv b^{p^i}$ mod $p^{i+1}$. All possible $\sigma(a_0)$ are congruent mod $p$; we want something unique mod $p^{n+1}$, so why not write $\sigma(a_0)^{p^n}$ in the 0-th coordinate. The first coordinate will be multiplied by $p$ anyway, so we only have to raise it to the $p^{n-1}$-th power to make it unique mod $p^{n+1}$. Upshot:
$$\sigma(a_0)^{p^n} + \sigma(a_1)^{p^{n-1}} p + ... + \sigma(a_n) p^n \in A/p^{n+1} A$$
is independent of $\sigma$; in a way, it is a canonical representative in $A/p^{n+1} A$ of one element in $k^{n+1}$. And it is induced by the (non-naive) Witt polynomial
$$W(X_0, ..., X_n) = X_0^{p^n} + p X_1^{p^{n-1}} + ... + p^n X_n .$$
In other words: For fixed $(a_0, ..., a_n)$, whatever liftings $\sigma_i$ one might choose, evaluating the Witt polynomial at $X_i = \sigma_i(a_i)$ will give the same element in $A/p^{n+1} A$. So it looks like it might produce universal formulae for a ring structure. (Still, it is astounding that these turn out to be polynomials.)
Finally, if $k$ is perfect, this coordinate map is still bijective, and we can and will normalise it. Depending on whether one considers the normalisation
$(a_0, ..., a_n) \mapsto (a_0^{p^{-n}}, ..., a_n^{p^{-n}})$ or
$(a_0, ..., a_n) \mapsto (a_0^{p^{-n}}, a_1^{p^{1-n}}, ..., a_n)$
to be more natural, in the limit one will look at the element
$\sum_{i=0}^\infty \tau (a_i^{p^{-i}}) p^i$ or
$\sum_{i=0}^\infty \tau (a_i) p^i$
as a natural representative in $A$ of the coordinates $(a_0, a_1, ...)$ -- where the map $\tau: a \mapsto \lim \sigma (a^{p^{-i}})^{p^i}$ is independent of $\sigma$ and turns out to be the Teichmüller map, which we have thus generalised by forgetting it for a while.
History: The relevant papers are
- Hasse, F. K. Schmidt: Die Struktur diskret bewerteter Körper. Crelle 170 (1934)
- H. L. Schmid: Zyklische algebraische Funktionenkörper vom Grad $p^n$ über endlichem Konstantenkörper der Charakteristik $p$. Crelle 175 (1936); received 6-I-1936
- Teichmüller: Über die Struktur diskret bewerteter Körper. Nachr. Ges. Wiss. Göttingen, 1936; received 21-II-1936
- Witt: Zyklische Körper und Algebren der Charakteristik $p$ vom Grad $p^n$. Struktur diskret bewerteter Körper mit vollkommenem Restklassenkörper der Charakteristik $p$, Crelle 176 (1937), dated 22-VI-1936, received 29-VIII-1936
- Teichmüller: Diskret bewertete perfekte Körper mit unvollkommenem Restklassenkörper, Crelle 176 (1937), received 5-IX-1936
(Beware, obsolete notation: "perfekt" $\sim$ complete; "(un)vollkommen" = (im)perfect)
In (1), a structure theory of complete discretely valued fields had already been done (!), although more complicated. As olli_jvn has already said, Witt was mainly working on generalising Artin-Schreier theory to what is now Artin-Schreier-Witt theory as well as constructing cyclic algebras of degree $p^n$. This is also what Schmid did in (2). This paper was discussed in an Arbeitsgemeinschaft led by Witt with participants Hasse, Teichmüller, Schmid and others. On the first page of (2), there is a note added during correction that Witt has found a "neues Kalkül" which simplifies Schmid's results. -- Hazewinkel notes (p. 5) that the Witt polynomials turn up in (3) and suggests that this might have inspired Witt, however if one looks where they come from here, one reads (p. 155 = p. 57 in Teichmüller's Collected Works):
"Tatsächlich ergibt sich das Verfahren
aus einem Formalismus, den H. L.
Schmid und E. Witt zu ganz anderen
Zwecken aufgestellt haben."
and then the Witt polynomials appear, and the summation polynomials (at least, mod p) are deduced from them. On the first page of (3), Teichmüller writes that this work was inspired by the mentioned Arbeitsgemeinschaft. In the introductions to (4) and (5), Witt and Teichmüller credit each other with realising the use of the "neues Kalkül" for the structure theory of complete discretely valued fields in the unequal characteristic case. As Hazewinkel writes (p. 9, in accordance with Witt's introduction in (4)), a decisive inspiration for Witt had been the "summation" polynomials that occured in (2), which are constructed recursively (in building an algebra of degree $p^n$ recursively by adding Artin-Schreier-like $p$-layers), and which for $n = 1$ reduce to a plain sum. Indeed, on p. 111 of (2), there are polynomials $z_\nu$ which in today's notation would be Witt's $S_\nu - X_\nu - Y_\nu$, defined recursively with the help of a polynomial $f_\nu$ to be found on p. 112, which is nothing else than the Witt polynomial $W_\nu$ in slightly different normalization. So presumably the timeline is:
Schmid presents his paper in the Arbeitsgemeinschaft (before January 1936) $\rightarrow$ Witt finds general Witt vector "Kalkül" (January 1936) $\rightarrow$ Witt and Teichmüller independently realise that this gives a structure theory of complete discretely valued fields with perfect residue field; Teichmüller finds sum and product polynomials (mod p) as well as Teichmüller representatives and reduction of the general case to the case of perfect residue field (January-February 1936) $\rightarrow$ Witt works out his whole theory, Witt and Teichmüller agree to put the perfect case among all the other applications into (4), a detailed treatment of the imperfect case in (5).
Best Answer
You may find the following more transparent, since it uses only the fact that the Witt vectors are a complete DVR with residue field $k$. Call the Witt vectors $R$, and let $y$ be a unit for which you want to find $z$ with $z^\sigma=yz$. First do it mod $p$, by solving $\zeta^p=\eta\zeta$ for $\zeta$ in $k$, where $\eta$ is the image of $y$ in $k$. Now you can assume that you have $z\in R$ satisfying $z^\sigma\equiv yz \mod{(p^m)}$, in other words $z^\sigma \equiv yz + p^m\delta \mod{(p^{m+1})}$. Now you want to adjust $z$ to $z'=z+p^m x$ so that $z'$ satisfies your congruence modulo $(p^{m+1})$. This boils down to solving $\xi^p - \xi \eta + \delta = 0$ in $k$, which you can do. So you see that you don't need $k$ to be algebraically closed, just separably closed.