Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, 1), (1, 0), (-1, -2), (-2, -1)$. This example came out from a numerical search followed by a rational approximation in order to obtain a proven result (thus eliminating possible rounding issues). The following Sage code verifies the example algebraically:

    R.<x, y> = QQ[]
    f = (2*x^4 - 15*x^3*y + 42*x^2*y^2 - 15*x*y^3 + 2*y^4 - 4*x^3 +
         18*x^2*y + 18*x*y^2 - 4*y^3 + 2*x^2 - 3*x*y + 2*y^2)
    fx, fy = f.derivative(x), f.derivative(y)
    critical_points = ideal([fx, fy]).variety()
    hesse = matrix(2, 2, [fx.derivative(x), fx.derivative(y),
                          fy.derivative(x), fy.derivative(y)])
    minima = []
    for cp in critical_points:
        h = hesse.substitute(cp)
        if h.trace() > 0 and h.det() > 0:
            minima.append(cp)
    assert len(minima) == 5

An additional nice feature of this example is its symmetry $f(x, y)=f(y, x)$.

*Remark*: Theorem 3.1.6. of the paper [Critical points of real polynomials ...][1] 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://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