**Recent addition:** Inspired by the comments by DimaPasechnik and Matt F. about sum of squares decompositions, I tried the following very natural idea: Try to find $f$ of the form $f(x,y)=A(x,y)^2+B(x,y)^2$, where $A$ and $B$ are quadratic polynomials whose zero sets intersect in $4$ points, which then of course are local minima, and attempt to get a fifth local minimum. By a shift we may look for it in $(0,0)$. And indeed, it is not hard to arrange things which yields an example which can be checked almost effortlessly. One possibility is
\begin{align}
A &= 4y^2-1\\
B &= 9y^2 + 4y - x^2.
\end{align}
The curves $A=0$ and $B=0$ intersect in the four points
\begin{equation}
(\pm\frac{1}{2}, -\frac{1}{2}), (\pm\frac{\sqrt{17}}{2}, \frac{1}{2}).
\end{equation}
The partial derivatives of $A^2$ and $B^2$ clearly vanish in $(0,0)$, so $(0,0)$ is a critical point of $f=A^2+B^2$. The Hesse matrix however is only positive semidefinite. Nevertheless, $(0,0)$ is a local isolated minimum, because $f(x,y)=1+8y^2+$ terms of degree $\ge3$ and $f(x,0)=x^4+1$.

A similar example, where the Hessian in $(0,0)$ actually is positive definite, is $f=A^2+B^2$ for $A=x^2 + y - 2$ and $B = 3x^2 - 2y^2 + 2y + 1$. Note that $f=5+2x^2 + y^2+$ terms of degree $\ge3$.

**Original answer:** The quartic
\begin{equation}
f = 3x^4 - 5x^2y^2 + 5y^4 + 20x^2y - 32y^3 + x^2 + 2y^2,
\end{equation}
which is even in $x$, has $5$ isolated local minima in
\begin{equation}
(0, 0), (\pm2, -1), (\pm2, 5).
\end{equation}
In order to prove the claim, one checks that the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ vanish and that the Hessian is positive definite in these points, where the latter is equivalent to the traces and the determinants being positive.

If one carries out the calculation by hand, then the symmetry $f(x,y)=f(-x,y)$ requires only to check $3$ points. Alternatively, the following Sage code verifies the example:

    R.<x, y> = QQ[]
    f = 3*x^4-5*x^2*y^2+5*y^4+20*x^2*y-32*y^3+x^2+2*y^2
    fx, fy = f.derivative(x), f.derivative(y)
    fxx = fx.derivative(x)
    fxy = fx.derivative(y)
    fyy = fy.derivative(y)
    tra, det = fxx+fyy, fxx*fyy-fxy^2
    cps = ideal([fx, fy]).variety()
    minima = [p for p in cps if tra.subs(p) > 0 and det.subs(p) > 0]
    print(len(minima) == 5)

*Remark 1:* As pointed out by Jap88 in the comments (see also his nice visualizations at [https://math.stackexchange.com/questions/4620663][1]), previous examples are difficult to visualize. This holds true for this one too. The reason is that some of the saddles points tend to be very close to the minima. The red and blue lines are the curves given by $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$, and the green dots are the minima. As the first picture doesn't show well the situation around $(0,0)$, the second one zooms into this area:[![][2]][2]

*Remark 2:* Another rather short example is
\begin{equation}
f = x^4 - 3x^2y^2 + 3y^4 - 2x^2y + 3y^3 + x^2 - 2y^2.
\end{equation}
Here, the $5$ local minima are irrational, so the exact algebraic verification by hand is a little more involved. Or, if one relies on Sage again, then one simply replaces the first two lines of the code above with

    R.<x, y> = AA[]
    f = x^4 - 3*x^2*y^2 + 3*y^4 - 2*x^2*y + 3*y^3 + x^2 - 2*y^2

Here <tt>AA</tt> is the (exact) field of real algebraic numbers. The image (notation as above) is [![enter image description here][3]][3]

*Remark 3:* Conjecture 5.4 in [this paper by Durfee et.al][4] predicts that a quartic has at most $4$ isolated local minima. So this is a counterexample.

*Remark 4*: Theorem 3.1.6. of the paper [Critical points of real polynomials ...][5] by Shustin seems to claim that $5$ minima are possible. However, the "proof" given in Section 3.13 merely says "Assume that $d \le 4$. Then all the index distributions except [...] can be easily realized as it was explained above". I don't know what the author means by "above", as all the previous cases handle degrees $d\ge5$. 


  [1]: https://math.stackexchange.com/questions/4620663
  [2]: https://i.sstatic.net/Xux9G.png
  [3]: https://i.sstatic.net/0Lhl3.png
  [4]: http://www.jstor.org/stable/2324459
  [5]: https://www.researchgate.net/profile/Eugenii-Shustin-2/publication/267655438_Critical_points_of_real_polynomials_subdivisions_of_Newton_polyhedra_and_topology_of_real_algebraic_hypersurfaces/links/593ec9480f7e9bf167c5c86c/Critical-points-of-real-polynomials-subdivisions-of-Newton-polyhedra-and-topology-of-real-algebraic-hypersurfaces.pdf?origin=publicationDetail&_sg%5B0%5D=q9Oh1_m3jOK6GbcVqAfUCbUZQcUhzBdCSuiy3C4-M4tptZBRj9TUFvwMlxungIYZGBJCN448_Ym-sIGtnLa7Tg.4tpQfI3YawFUZmb-f-azz2eQ4vhkYXKkq72yY3dxP10EjQoQVfzrjiDaqocMD5tauPGPA_EHk65Awgyzqi5H6A&_sg%5B1%5D=SGk8BZqoH6nDDCqLNnHvxuM-t9dfzYMqDyjU0p1TCYgtjQarC-uAAejb8-TZRRJc3tDQhpv1GlfrHxvmiaWwho-UDx5azx9mjQ7Oto0W3pHr.4tpQfI3YawFUZmb-f-azz2eQ4vhkYXKkq72yY3dxP10EjQoQVfzrjiDaqocMD5tauPGPA_EHk65Awgyzqi5H6A&_iepl=&_rtd=eyJjb250ZW50SW50ZW50IjoibWFpbkl0ZW0ifQ%3D%3D