Yes. In fact there is a clean Hodge–theoretic criterion, and it gives a positive answer (indeed, in much greater generality).

Let V = H^1_B(E, Q) be the rational Betti Hodge structure of weight 1 attached to an elliptic curve E over a number field K. Over C one has the Hodge decomposition V ⊗ C = V^{1,0} ⊕ V^{0,1}, and over K one has the de Rham exact sequence 0 → H^0(E, Ω^1_{E/K}) → H^1_dR(E/K) → H^1(E, O_E) → 0. Your hypothesis says that, after choosing an embedding K ↪ C and using the de Rham–Betti comparison, the Hodge decomposition of H^1_dR(E_C/C) is induced by a splitting over K of the de Rham sequence. Equivalently, the Hodge filtration on H^1 is defined over a number field.

General Hodge–theoretic fact. For a polarizable Q–Hodge structure V of type {(-1,0),(0,-1)} (i.e., coming from an abelian variety), the following are equivalent:

• The Mumford–Tate group MT(V) is a torus (i.e., V is of CM type).
• The Hodge filtration on V ⊗ C is defined over Q̄ (equivalently, there is a number field L and an L-splitting whose base change to C is the Hodge decomposition).

References for this equivalence include Deligne, Hodge cycles on abelian varieties (in Hodge Cycles, Motives, and Shimura Varieties, LNM 900), and standard expositions such as Milne’s “Complex Multiplication” notes (see e.g. the discussion of Mumford–Tate groups and CM type).

Applying this to V = H^1_B(E, Q), one obtains:

• If the Hodge decomposition on H^1_dR(E_C/C) is induced by a splitting over a number field (as in your hypothesis), then MT(H^1(E)) is a torus, hence E is of CM type. For an elliptic curve, this simply means End^0(E) is an imaginary quadratic field.

There is also a classical analytic proof in the elliptic-curve case using periods and Schneider’s theorem: the existence of a splitting defined over a number field forces the ratio of periods τ to be algebraic; since j(E) is algebraic, Schneider’s theorem then implies τ is imaginary quadratic, i.e., E has CM. But the Mumford–Tate argument above is shorter and works uniformly for all abelian varieties.

So the converse Katz asked about is true: an elliptic curve whose Hodge decomposition is induced by a splitting of the algebraic de Rham cohomology necessarily has complex multiplication.