Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, 82), (82, 0), (-253, 593), (-649, 237)$. 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 = (46324004878805168630254223262455*x^4 +
         306992632289131400429573117563212*x^3*y +
         634038569228212054158919111315662*x^2*y^2 +
         413369023506452188951928070902676*x*y^3 +
         83898898446890528970856409397843*y^4 -
         5065324255328303287915841363261976*x^3 -
         25173395847708774835224995640183384*x^2*y -
         33896259927529079494058101814019432*x*y^2 -
         11038630209215743721318102045519592*y^3 +
         69665795209396673989693247728208*x^2 +
         229479129419752644122049558016717152*y^2)
    fx = f.derivative(x)
    fy = f.derivative(y)
    critical_points = ideal([fx, fy]).variety()
    assert len(critical_points) == 5
    hesse = matrix(2, 2, [fx.derivative(x), fx.derivative(y),
                          fy.derivative(x), fy.derivative(y)])
    for cp in critical_points:
        h = hesse.substitute(cp)
        assert h.trace() > 0
        assert h.det() > 0

*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