We introduce the notion of complex chromatic number of signed graphs as follows: given the set \({\mathbb {C}}_{k,l}=\{\pm 1, \pm 2, \ldots , \pm k\}\cup \{\pm 1i, \pm 2i, \ldots , \pm li\}\), where \(i=\sqrt{-1}\), a signed graph \((G,\sigma )\) is said to be (k, l)-colorable if there exists a mapping c of vertices of G to \({\mathbb {C}}_{k,l}\) such that for every edge xy of G we have $$\begin{aligned} c(x)c(y)\ne \sigma (xy) |c(x)^2|. \end{aligned}$$The complex chromatic number of a signed graph \((G, \sigma )\), denoted \(\chi _{com}(G, \sigma )\), is defined to be the smallest order of \({\mathbb {C}}_{k,l}\) such that \((G, \sigma )\) admits a (k, l)-coloring. In this work, after providing an equivalent definition in the language of homomorphisms of signed graphs, we show that there are signed planar simple graphs which are not 4-colorable. That is to say: there is a signed planar simple graph which is neither (2, 0)-colorable, nor (1, 1)-colorable, nor (0, 2)-colorable. That every signed planar simple graph is (2, 0)-colorable was the subject of a conjecture by Máçajová, Raspaud and Škoviera which was recently disproved by Kardoš and Narboni using a dual notion. We provide a direct approach and a short proof. That every signed planar simple graph is (1, 1)-colorable is a recent conjecture of Jiang and Zhu which we disprove in this work. Noting that (0, 2)-coloring of \((G,\sigma )\) is the same as (2, 0)-coloring of \((G, -\sigma )\), this proves the existence of a signed planar simple graph whose complex chromatic number is larger than 4. Further developing the homomorphism approach, and as an analogue of the 5-color theorem, we find three minimal signed graphs each on three vertices, without a \(K_1^{\pm }\) (a vertex with both a positive and a negative loop) and each having the property of admitting a homomorphism from every signed planar simple graph. Finally we identify several other problems of high interest in colorings and homomorphisms of signed planar simple graphs.