Can an electrocardiogram be recorded with a single electrode?
Short answer
Any electrophysiological recording of potentials, including the electrocardiogram (ECG or EKG), measures a potential difference. Hence, two or more electrodes are necessary.

Background
The ECG measures the rhythmic voltages changes generated by the heart muscle. Voltage is defined as a potential difference between two points. Therefore, a voltage is always measured across two or more points. For example, when
you measure voltage when holding one out of two recording electrodes to one of the two poles of a battery, nothing will be recorded. This, because no current will flow through the potentiometer by connecting just one pole of the battery. Likewise, electrophysiology needs at least two recording channels to determine the voltage differential between the two. One channel is typically called the active electrode, the other the reference. The configuration of active and reference determines the polarity of the recorded voltage. Often a ground electrode is added to remove background noise.